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Gleichwertig: ",[573,835,837,850],{"className":836},[576],[573,838,840],{"className":839},[580],[582,841,842],{"xmlns":584},[586,843,844,848],{},[589,845,846],{},[592,847,633],{},[596,849,633],{"encoding":598},[573,851,853],{"className":852,"ariaHidden":603},[602],[573,854,856,859],{"className":855},[607],[573,857],{"className":858,"style":612},[611],[573,860,633],{"className":861},[616,617]," und ",[573,864,866,879],{"className":865},[576],[573,867,869],{"className":868},[580],[582,870,871],{"xmlns":584},[586,872,873,877],{},[589,874,875],{},[592,876,640],{},[596,878,640],{"encoding":598},[573,880,882],{"className":881,"ariaHidden":603},[602],[573,883,885,888],{"className":884},[607],[573,886],{"className":887,"style":703},[611],[573,889,640],{"className":890},[616,617]," lassen bei Division durch ",[573,893,895,908],{"className":894},[576],[573,896,898],{"className":897},[580],[582,899,900],{"xmlns":584},[586,901,902,906],{},[589,903,904],{},[592,905,594],{},[596,907,594],{"encoding":598},[573,909,911],{"className":910,"ariaHidden":603},[602],[573,912,914,917],{"className":913},[607],[573,915],{"className":916,"style":612},[611],[573,918,594],{"className":919},[616,617]," denselben Rest. Die Zahlen mit gleichem Rest bilden eine ",[569,922,923],{},"Restklasse",".",[565,926,927,928,956,957,1147,1148,1205],{},"Die Kongruenz verträgt sich mit Addition, Subtraktion und Multiplikation: Man darf jede Zahl durch einen Vertreter mit gleichem Rest ersetzen, ohne dass sich das Ergebnis modulo ",[573,929,931,944],{"className":930},[576],[573,932,934],{"className":933},[580],[582,935,936],{"xmlns":584},[586,937,938,942],{},[589,939,940],{},[592,941,594],{},[596,943,594],{"encoding":598},[573,945,947],{"className":946,"ariaHidden":603},[602],[573,948,950,953],{"className":949},[607],[573,951],{"className":952,"style":612},[611],[573,954,594],{"className":955},[616,617]," ändert. Daraus folgt auch ",[573,958,960,1013],{"className":959},[576],[573,961,963],{"className":962},[580],[582,964,965],{"xmlns":584},[586,966,967,1010],{},[589,968,969,977,979,990,992,994,996,1004,1006,1008],{},[970,971,972,974],"msup",{},[592,973,633],{},[592,975,976],{},"k",[635,978,637],{},[970,980,981,983],{},[592,982,633],{},[589,984,985,988],{},[635,986,987],{"mathvariant":655},"′",[592,989,976],{},[642,991],{},[642,993],{"width":646},[635,995,650],{"stretchy":649},[589,997,998,1000,1002],{},[592,999,594],{"mathvariant":655},[592,1001,658],{"mathvariant":655},[592,1003,661],{"mathvariant":655},[642,1005],{"width":664},[592,1007,594],{},[635,1009,669],{"stretchy":649},[596,1011,1012],{"encoding":598},"a^k \\equiv a'^k \\pmod m",[573,1014,1016,1073,1120],{"className":1015,"ariaHidden":603},[602],[573,1017,1019,1023,1064,1067,1070],{"className":1018},[607],[573,1020],{"className":1021,"style":1022},[611],"height:0.8491em;",[573,1024,1026,1029],{"className":1025},[616],[573,1027,633],{"className":1028},[616,617],[573,1030,1033],{"className":1031},[1032],"msupsub",[573,1034,1037],{"className":1035},[1036],"vlist-t",[573,1038,1041],{"className":1039},[1040],"vlist-r",[573,1042,1045],{"className":1043,"style":1022},[1044],"vlist",[573,1046,1048,1053],{"style":1047},"top:-3.063em;margin-right:0.05em;",[573,1049],{"className":1050,"style":1052},[1051],"pstrut","height:2.7em;",[573,1054,1060],{"className":1055},[1056,1057,1058,1059],"sizing","reset-size6","size3","mtight",[573,1061,976],{"className":1062,"style":1063},[616,617,1059],"margin-right:0.0315em;",[573,1065],{"className":1066,"style":689},[642],[573,1068,637],{"className":1069},[693],[573,1071],{"className":1072,"style":689},[642],[573,1074,1076,1079,1114,1117],{"className":1075},[607],[573,1077],{"className":1078,"style":1022},[611],[573,1080,1082,1085],{"className":1081},[616],[573,1083,633],{"className":1084},[616,617],[573,1086,1088],{"className":1087},[1032],[573,1089,1091],{"className":1090},[1036],[573,1092,1094],{"className":1093},[1040],[573,1095,1097],{"className":1096,"style":1022},[1044],[573,1098,1099,1102],{"style":1047},[573,1100],{"className":1101,"style":1052},[1051],[573,1103,1105],{"className":1104},[1056,1057,1058,1059],[573,1106,1108,1111],{"className":1107},[616,1059],[573,1109,987],{"className":1110},[616,1059],[573,1112,976],{"className":1113,"style":1063},[616,617,1059],[573,1115],{"className":1116},[642,710],[573,1118],{"className":1119,"style":714},[642],[573,1121,1123,1126,1129,1138,1141,1144],{"className":1122},[607],[573,1124],{"className":1125,"style":721},[611],[573,1127,650],{"className":1128},[725],[573,1130,1132],{"className":1131},[616],[573,1133,1135],{"className":1134},[616],[573,1136,736],{"className":1137},[616,735],[573,1139],{"className":1140,"style":740},[642],[573,1142,594],{"className":1143},[616,617],[573,1145,669],{"className":1146},[747]," für jedes ",[573,1149,1151,1172],{"className":1150},[576],[573,1152,1154],{"className":1153},[580],[582,1155,1156],{"xmlns":584},[586,1157,1158,1169],{},[589,1159,1160,1162,1165],{},[592,1161,976],{},[635,1163,1164],{},"∈",[592,1166,1168],{"mathvariant":1167},"double-struck","N",[596,1170,1171],{"encoding":598},"k \\in \\mathbb{N}",[573,1173,1175,1194],{"className":1174,"ariaHidden":603},[602],[573,1176,1178,1182,1185,1188,1191],{"className":1177},[607],[573,1179],{"className":1180,"style":1181},[611],"height:0.7335em;vertical-align:-0.0391em;",[573,1183,976],{"className":1184,"style":1063},[616,617],[573,1186],{"className":1187,"style":689},[642],[573,1189,1164],{"className":1190},[693],[573,1192],{"className":1193,"style":689},[642],[573,1195,1197,1201],{"className":1196},[607],[573,1198],{"className":1199,"style":1200},[611],"height:0.6889em;",[573,1202,1168],{"className":1203},[616,1204],"mathbb",". In der Praxis reduziert man deshalb nach jedem Rechenschritt, damit die Zahlen klein bleiben.",[565,1207,1208,1211,1212,1363,1364,1467,1468,1562,1563,1591,1592,1620],{},[569,1209,1210],{},"Division"," ist nur eingeschränkt erlaubt: Aus ",[573,1213,1215,1263],{"className":1214},[576],[573,1216,1218],{"className":1217},[580],[582,1219,1220],{"xmlns":584},[586,1221,1222,1260],{},[589,1223,1224,1227,1230,1232,1234,1236,1238,1240,1242,1244,1246,1254,1256,1258],{},[592,1225,1226],{},"c",[635,1228,1229],{},"⋅",[592,1231,633],{},[635,1233,637],{},[592,1235,1226],{},[635,1237,1229],{},[592,1239,640],{},[642,1241],{},[642,1243],{"width":646},[635,1245,650],{"stretchy":649},[589,1247,1248,1250,1252],{},[592,1249,594],{"mathvariant":655},[592,1251,658],{"mathvariant":655},[592,1253,661],{"mathvariant":655},[642,1255],{"width":664},[592,1257,594],{},[635,1259,669],{"stretchy":649},[596,1261,1262],{"encoding":598},"c \\cdot a \\equiv c \\cdot b \\pmod m",[573,1264,1266,1285,1303,1321,1336],{"className":1265,"ariaHidden":603},[602],[573,1267,1269,1273,1276,1279,1282],{"className":1268},[607],[573,1270],{"className":1271,"style":1272},[611],"height:0.4445em;",[573,1274,1226],{"className":1275},[616,617],[573,1277],{"className":1278,"style":816},[642],[573,1280,1229],{"className":1281},[820],[573,1283],{"className":1284,"style":816},[642],[573,1286,1288,1291,1294,1297,1300],{"className":1287},[607],[573,1289],{"className":1290,"style":682},[611],[573,1292,633],{"className":1293},[616,617],[573,1295],{"className":1296,"style":689},[642],[573,1298,637],{"className":1299},[693],[573,1301],{"className":1302,"style":689},[642],[573,1304,1306,1309,1312,1315,1318],{"className":1305},[607],[573,1307],{"className":1308,"style":1272},[611],[573,1310,1226],{"className":1311},[616,617],[573,1313],{"className":1314,"style":816},[642],[573,1316,1229],{"className":1317},[820],[573,1319],{"className":1320,"style":816},[642],[573,1322,1324,1327,1330,1333],{"className":1323},[607],[573,1325],{"className":1326,"style":703},[611],[573,1328,640],{"className":1329},[616,617],[573,1331],{"className":1332},[642,710],[573,1334],{"className":1335,"style":714},[642],[573,1337,1339,1342,1345,1354,1357,1360],{"className":1338},[607],[573,1340],{"className":1341,"style":721},[611],[573,1343,650],{"className":1344},[725],[573,1346,1348],{"className":1347},[616],[573,1349,1351],{"className":1350},[616],[573,1352,736],{"className":1353},[616,735],[573,1355],{"className":1356,"style":740},[642],[573,1358,594],{"className":1359},[616,617],[573,1361,669],{"className":1362},[747]," folgt ",[573,1365,1367,1404],{"className":1366},[576],[573,1368,1370],{"className":1369},[580],[582,1371,1372],{"xmlns":584},[586,1373,1374,1402],{},[589,1375,1376,1378,1380,1382,1384,1386,1388,1396,1398,1400],{},[592,1377,633],{},[635,1379,637],{},[592,1381,640],{},[642,1383],{},[642,1385],{"width":646},[635,1387,650],{"stretchy":649},[589,1389,1390,1392,1394],{},[592,1391,594],{"mathvariant":655},[592,1393,658],{"mathvariant":655},[592,1395,661],{"mathvariant":655},[642,1397],{"width":664},[592,1399,594],{},[635,1401,669],{"stretchy":649},[596,1403,672],{"encoding":598},[573,1405,1407,1425,1440],{"className":1406,"ariaHidden":603},[602],[573,1408,1410,1413,1416,1419,1422],{"className":1409},[607],[573,1411],{"className":1412,"style":682},[611],[573,1414,633],{"className":1415},[616,617],[573,1417],{"className":1418,"style":689},[642],[573,1420,637],{"className":1421},[693],[573,1423],{"className":1424,"style":689},[642],[573,1426,1428,1431,1434,1437],{"className":1427},[607],[573,1429],{"className":1430,"style":703},[611],[573,1432,640],{"className":1433},[616,617],[573,1435],{"className":1436},[642,710],[573,1438],{"className":1439,"style":714},[642],[573,1441,1443,1446,1449,1458,1461,1464],{"className":1442},[607],[573,1444],{"className":1445,"style":721},[611],[573,1447,650],{"className":1448},[725],[573,1450,1452],{"className":1451},[616],[573,1453,1455],{"className":1454},[616],[573,1456,736],{"className":1457},[616,735],[573,1459],{"className":1460,"style":740},[642],[573,1462,594],{"className":1463},[616,617],[573,1465,669],{"className":1466},[747]," nur, wenn ",[573,1469,1471,1507],{"className":1470},[576],[573,1472,1474],{"className":1473},[580],[582,1475,1476],{"xmlns":584},[586,1477,1478,1504],{},[589,1479,1480,1483,1486,1488,1490,1493,1495,1497,1500],{},[592,1481,1482],{"mathvariant":655},"ggT",[635,1484,1485],{},"⁡",[635,1487,650],{"stretchy":649},[592,1489,1226],{},[635,1491,1492],{"separator":603},",",[592,1494,594],{},[635,1496,669],{"stretchy":649},[635,1498,1499],{},"=",[1501,1502,1503],"mn",{},"1",[596,1505,1506],{"encoding":598},"\\operatorname{ggT}(c, m) = 1",[573,1508,1510,1552],{"className":1509,"ariaHidden":603},[602],[573,1511,1513,1516,1523,1526,1529,1533,1537,1540,1543,1546,1549],{"className":1512},[607],[573,1514],{"className":1515,"style":721},[611],[573,1517,1520],{"className":1518},[1519],"mop",[573,1521,1482],{"className":1522},[616,735],[573,1524,650],{"className":1525},[725],[573,1527,1226],{"className":1528},[616,617],[573,1530,1492],{"className":1531},[1532],"mpunct",[573,1534],{"className":1535,"style":1536},[642],"margin-right:0.1667em;",[573,1538,594],{"className":1539},[616,617],[573,1541,669],{"className":1542},[747],[573,1544],{"className":1545,"style":689},[642],[573,1547,1499],{"className":1548},[693],[573,1550],{"className":1551,"style":689},[642],[573,1553,1555,1559],{"className":1554},[607],[573,1556],{"className":1557,"style":1558},[611],"height:0.6444em;",[573,1560,1503],{"className":1561},[616],". Dann gibt es ein Inverses von ",[573,1564,1566,1579],{"className":1565},[576],[573,1567,1569],{"className":1568},[580],[582,1570,1571],{"xmlns":584},[586,1572,1573,1577],{},[589,1574,1575],{},[592,1576,1226],{},[596,1578,1226],{"encoding":598},[573,1580,1582],{"className":1581,"ariaHidden":603},[602],[573,1583,1585,1588],{"className":1584},[607],[573,1586],{"className":1587,"style":612},[611],[573,1589,1226],{"className":1590},[616,617]," modulo ",[573,1593,1595,1608],{"className":1594},[576],[573,1596,1598],{"className":1597},[580],[582,1599,1600],{"xmlns":584},[586,1601,1602,1606],{},[589,1603,1604],{},[592,1605,594],{},[596,1607,594],{"encoding":598},[573,1609,1611],{"className":1610,"ariaHidden":603},[602],[573,1612,1614,1617],{"className":1613},[607],[573,1615],{"className":1616,"style":612},[611],[573,1618,594],{"className":1619},[616,617],", mit dem man multiplizieren kann.",[560,1622,1624],{"id":1623},"beispiele","Beispiele",[565,1626,1627,1630,1631,1684,1685,1714,1715,862,1767,1871],{},[569,1628,1629],{},"Neunerprobe:"," Ist ",[573,1632,1634,1654],{"className":1633},[576],[573,1635,1637],{"className":1636},[580],[582,1638,1639],{"xmlns":584},[586,1640,1641,1651],{},[589,1642,1643,1646,1648],{},[1501,1644,1645],{},"123",[635,1647,1229],{},[1501,1649,1650],{},"456",[596,1652,1653],{"encoding":598},"123 \\cdot 456",[573,1655,1657,1675],{"className":1656,"ariaHidden":603},[602],[573,1658,1660,1663,1666,1669,1672],{"className":1659},[607],[573,1661],{"className":1662,"style":1558},[611],[573,1664,1645],{"className":1665},[616],[573,1667],{"className":1668,"style":816},[642],[573,1670,1229],{"className":1671},[820],[573,1673],{"className":1674,"style":816},[642],[573,1676,1678,1681],{"className":1677},[607],[573,1679],{"className":1680,"style":1558},[611],[573,1682,1650],{"className":1683},[616]," durch ",[573,1686,1688,1702],{"className":1687},[576],[573,1689,1691],{"className":1690},[580],[582,1692,1693],{"xmlns":584},[586,1694,1695,1700],{},[589,1696,1697],{},[1501,1698,1699],{},"9",[596,1701,1699],{"encoding":598},[573,1703,1705],{"className":1704,"ariaHidden":603},[602],[573,1706,1708,1711],{"className":1707},[607],[573,1709],{"className":1710,"style":1558},[611],[573,1712,1699],{"className":1713},[616]," teilbar? Es gilt ",[573,1716,1718,1737],{"className":1717},[576],[573,1719,1721],{"className":1720},[580],[582,1722,1723],{"xmlns":584},[586,1724,1725,1734],{},[589,1726,1727,1729,1731],{},[1501,1728,1645],{},[635,1730,637],{},[1501,1732,1733],{},"6",[596,1735,1736],{"encoding":598},"123 \\equiv 6",[573,1738,1740,1758],{"className":1739,"ariaHidden":603},[602],[573,1741,1743,1746,1749,1752,1755],{"className":1742},[607],[573,1744],{"className":1745,"style":1558},[611],[573,1747,1645],{"className":1748},[616],[573,1750],{"className":1751,"style":689},[642],[573,1753,637],{"className":1754},[693],[573,1756],{"className":1757,"style":689},[642],[573,1759,1761,1764],{"className":1760},[607],[573,1762],{"className":1763,"style":1558},[611],[573,1765,1733],{"className":1766},[616],[573,1768,1770,1808],{"className":1769},[576],[573,1771,1773],{"className":1772},[580],[582,1774,1775],{"xmlns":584},[586,1776,1777,1805],{},[589,1778,1779,1781,1783,1785,1787,1789,1791,1799,1801,1803],{},[1501,1780,1650],{},[635,1782,637],{},[1501,1784,1733],{},[642,1786],{},[642,1788],{"width":646},[635,1790,650],{"stretchy":649},[589,1792,1793,1795,1797],{},[592,1794,594],{"mathvariant":655},[592,1796,658],{"mathvariant":655},[592,1798,661],{"mathvariant":655},[642,1800],{"width":664},[1501,1802,1699],{},[635,1804,669],{"stretchy":649},[596,1806,1807],{"encoding":598},"456 \\equiv 6 \\pmod 9",[573,1809,1811,1829,1844],{"className":1810,"ariaHidden":603},[602],[573,1812,1814,1817,1820,1823,1826],{"className":1813},[607],[573,1815],{"className":1816,"style":1558},[611],[573,1818,1650],{"className":1819},[616],[573,1821],{"className":1822,"style":689},[642],[573,1824,637],{"className":1825},[693],[573,1827],{"className":1828,"style":689},[642],[573,1830,1832,1835,1838,1841],{"className":1831},[607],[573,1833],{"className":1834,"style":1558},[611],[573,1836,1733],{"className":1837},[616],[573,1839],{"className":1840},[642,710],[573,1842],{"className":1843,"style":714},[642],[573,1845,1847,1850,1853,1862,1865,1868],{"className":1846},[607],[573,1848],{"className":1849,"style":721},[611],[573,1851,650],{"className":1852},[725],[573,1854,1856],{"className":1855},[616],[573,1857,1859],{"className":1858},[616],[573,1860,736],{"className":1861},[616,735],[573,1863],{"className":1864,"style":740},[642],[573,1866,1699],{"className":1867},[616],[573,1869,669],{"className":1870},[747],", also",[573,1873,1876],{"className":1874},[1875],"katex-display",[573,1877,1879,1937],{"className":1878},[576],[573,1880,1882],{"className":1881},[580],[582,1883,1885],{"xmlns":584,"display":1884},"block",[586,1886,1887,1934],{},[589,1888,1889,1891,1893,1895,1897,1899,1901,1903,1905,1908,1910,1913,1915,1918,1920,1928,1930,1932],{},[1501,1890,1645],{},[635,1892,1229],{},[1501,1894,1650],{},[635,1896,637],{},[1501,1898,1733],{},[635,1900,1229],{},[1501,1902,1733],{},[635,1904,1499],{},[1501,1906,1907],{},"36",[635,1909,637],{},[1501,1911,1912],{},"0",[642,1914],{},[642,1916],{"width":1917},"1em",[635,1919,650],{"stretchy":649},[589,1921,1922,1924,1926],{},[592,1923,594],{"mathvariant":655},[592,1925,658],{"mathvariant":655},[592,1927,661],{"mathvariant":655},[642,1929],{"width":664},[1501,1931,1699],{},[635,1933,669],{"stretchy":649},[596,1935,1936],{"encoding":598},"123 \\cdot 456 \\equiv 6 \\cdot 6 = 36 \\equiv 0 \\pmod 9",[573,1938,1940,1958,1976,1994,2012,2030,2046],{"className":1939,"ariaHidden":603},[602],[573,1941,1943,1946,1949,1952,1955],{"className":1942},[607],[573,1944],{"className":1945,"style":1558},[611],[573,1947,1645],{"className":1948},[616],[573,1950],{"className":1951,"style":816},[642],[573,1953,1229],{"className":1954},[820],[573,1956],{"className":1957,"style":816},[642],[573,1959,1961,1964,1967,1970,1973],{"className":1960},[607],[573,1962],{"className":1963,"style":1558},[611],[573,1965,1650],{"className":1966},[616],[573,1968],{"className":1969,"style":689},[642],[573,1971,637],{"className":1972},[693],[573,1974],{"className":1975,"style":689},[642],[573,1977,1979,1982,1985,1988,1991],{"className":1978},[607],[573,1980],{"className":1981,"style":1558},[611],[573,1983,1733],{"className":1984},[616],[573,1986],{"className":1987,"style":816},[642],[573,1989,1229],{"className":1990},[820],[573,1992],{"className":1993,"style":816},[642],[573,1995,1997,2000,2003,2006,2009],{"className":1996},[607],[573,1998],{"className":1999,"style":1558},[611],[573,2001,1733],{"className":2002},[616],[573,2004],{"className":2005,"style":689},[642],[573,2007,1499],{"className":2008},[693],[573,2010],{"className":2011,"style":689},[642],[573,2013,2015,2018,2021,2024,2027],{"className":2014},[607],[573,2016],{"className":2017,"style":1558},[611],[573,2019,1907],{"className":2020},[616],[573,2022],{"className":2023,"style":689},[642],[573,2025,637],{"className":2026},[693],[573,2028],{"className":2029,"style":689},[642],[573,2031,2033,2036,2039,2042],{"className":2032},[607],[573,2034],{"className":2035,"style":1558},[611],[573,2037,1912],{"className":2038},[616],[573,2040],{"className":2041},[642,710],[573,2043],{"className":2044,"style":2045},[642],"margin-right:1em;",[573,2047,2049,2052,2055,2064,2067,2070],{"className":2048},[607],[573,2050],{"className":2051,"style":721},[611],[573,2053,650],{"className":2054},[725],[573,2056,2058],{"className":2057},[616],[573,2059,2061],{"className":2060},[616],[573,2062,736],{"className":2063},[616,735],[573,2065],{"className":2066,"style":740},[642],[573,2068,1699],{"className":2069},[616],[573,2071,669],{"className":2072},[747],[565,2074,2075,2076,2163,2164,2193,2194,2222],{},"Tatsächlich ist ",[573,2077,2079,2109],{"className":2078},[576],[573,2080,2082],{"className":2081},[580],[582,2083,2084],{"xmlns":584},[586,2085,2086,2106],{},[589,2087,2088,2090,2092,2094,2096,2099,2103],{},[1501,2089,1645],{},[635,2091,1229],{},[1501,2093,1650],{},[635,2095,1499],{},[1501,2097,2098],{},"56",[2100,2101,2102],"mtext",{}," ",[1501,2104,2105],{},"088",[596,2107,2108],{"encoding":598},"123 \\cdot 456 = 56\\,088",[573,2110,2112,2130,2148],{"className":2111,"ariaHidden":603},[602],[573,2113,2115,2118,2121,2124,2127],{"className":2114},[607],[573,2116],{"className":2117,"style":1558},[611],[573,2119,1645],{"className":2120},[616],[573,2122],{"className":2123,"style":816},[642],[573,2125,1229],{"className":2126},[820],[573,2128],{"className":2129,"style":816},[642],[573,2131,2133,2136,2139,2142,2145],{"className":2132},[607],[573,2134],{"className":2135,"style":1558},[611],[573,2137,1650],{"className":2138},[616],[573,2140],{"className":2141,"style":689},[642],[573,2143,1499],{"className":2144},[693],[573,2146],{"className":2147,"style":689},[642],[573,2149,2151,2154,2157,2160],{"className":2150},[607],[573,2152],{"className":2153,"style":1558},[611],[573,2155,2098],{"className":2156},[616],[573,2158],{"className":2159,"style":1536},[642],[573,2161,2105],{"className":2162},[616],", und die Quersumme ",[573,2165,2167,2181],{"className":2166},[576],[573,2168,2170],{"className":2169},[580],[582,2171,2172],{"xmlns":584},[586,2173,2174,2179],{},[589,2175,2176],{},[1501,2177,2178],{},"27",[596,2180,2178],{"encoding":598},[573,2182,2184],{"className":2183,"ariaHidden":603},[602],[573,2185,2187,2190],{"className":2186},[607],[573,2188],{"className":2189,"style":1558},[611],[573,2191,2178],{"className":2192},[616]," ist durch ",[573,2195,2197,2210],{"className":2196},[576],[573,2198,2200],{"className":2199},[580],[582,2201,2202],{"xmlns":584},[586,2203,2204,2208],{},[589,2205,2206],{},[1501,2207,1699],{},[596,2209,1699],{"encoding":598},[573,2211,2213],{"className":2212,"ariaHidden":603},[602],[573,2214,2216,2219],{"className":2215},[607],[573,2217],{"className":2218,"style":1558},[611],[573,2220,1699],{"className":2221},[616]," teilbar.",[565,2224,2225,2294,2295,2324,2325,2483],{},[569,2226,2227,2228,2293],{},"Letzte Ziffer von ",[573,2229,2231,2251],{"className":2230},[576],[573,2232,2234],{"className":2233},[580],[582,2235,2236],{"xmlns":584},[586,2237,2238,2248],{},[589,2239,2240],{},[970,2241,2242,2245],{},[1501,2243,2244],{},"7",[1501,2246,2247],{},"100",[596,2249,2250],{"encoding":598},"7^{100}",[573,2252,2254],{"className":2253,"ariaHidden":603},[602],[573,2255,2257,2261],{"className":2256},[607],[573,2258],{"className":2259,"style":2260},[611],"height:0.8141em;",[573,2262,2264,2267],{"className":2263},[616],[573,2265,2244],{"className":2266},[616],[573,2268,2270],{"className":2269},[1032],[573,2271,2273],{"className":2272},[1036],[573,2274,2276],{"className":2275},[1040],[573,2277,2279],{"className":2278,"style":2260},[1044],[573,2280,2281,2284],{"style":1047},[573,2282],{"className":2283,"style":1052},[1051],[573,2285,2287],{"className":2286},[1056,1057,1058,1059],[573,2288,2290],{"className":2289},[616,1059],[573,2291,2247],{"className":2292},[616,1059],":"," Gesucht ist der Rest modulo ",[573,2296,2298,2312],{"className":2297},[576],[573,2299,2301],{"className":2300},[580],[582,2302,2303],{"xmlns":584},[586,2304,2305,2310],{},[589,2306,2307],{},[1501,2308,2309],{},"10",[596,2311,2309],{"encoding":598},[573,2313,2315],{"className":2314,"ariaHidden":603},[602],[573,2316,2318,2321],{"className":2317},[607],[573,2319],{"className":2320,"style":1558},[611],[573,2322,2309],{"className":2323},[616],". Wegen ",[573,2326,2328,2376],{"className":2327},[576],[573,2329,2331],{"className":2330},[580],[582,2332,2333],{"xmlns":584},[586,2334,2335,2373],{},[589,2336,2337,2344,2346,2349,2351,2353,2355,2357,2359,2367,2369,2371],{},[970,2338,2339,2341],{},[1501,2340,2244],{},[1501,2342,2343],{},"4",[635,2345,1499],{},[1501,2347,2348],{},"2401",[635,2350,637],{},[1501,2352,1503],{},[642,2354],{},[642,2356],{"width":646},[635,2358,650],{"stretchy":649},[589,2360,2361,2363,2365],{},[592,2362,594],{"mathvariant":655},[592,2364,658],{"mathvariant":655},[592,2366,661],{"mathvariant":655},[642,2368],{"width":664},[1501,2370,2309],{},[635,2372,669],{"stretchy":649},[596,2374,2375],{"encoding":598},"7^4 = 2401 \\equiv 1 \\pmod{10}",[573,2377,2379,2423,2441,2456],{"className":2378,"ariaHidden":603},[602],[573,2380,2382,2385,2414,2417,2420],{"className":2381},[607],[573,2383],{"className":2384,"style":2260},[611],[573,2386,2388,2391],{"className":2387},[616],[573,2389,2244],{"className":2390},[616],[573,2392,2394],{"className":2393},[1032],[573,2395,2397],{"className":2396},[1036],[573,2398,2400],{"className":2399},[1040],[573,2401,2403],{"className":2402,"style":2260},[1044],[573,2404,2405,2408],{"style":1047},[573,2406],{"className":2407,"style":1052},[1051],[573,2409,2411],{"className":2410},[1056,1057,1058,1059],[573,2412,2343],{"className":2413},[616,1059],[573,2415],{"className":2416,"style":689},[642],[573,2418,1499],{"className":2419},[693],[573,2421],{"className":2422,"style":689},[642],[573,2424,2426,2429,2432,2435,2438],{"className":2425},[607],[573,2427],{"className":2428,"style":1558},[611],[573,2430,2348],{"className":2431},[616],[573,2433],{"className":2434,"style":689},[642],[573,2436,637],{"className":2437},[693],[573,2439],{"className":2440,"style":689},[642],[573,2442,2444,2447,2450,2453],{"className":2443},[607],[573,2445],{"className":2446,"style":1558},[611],[573,2448,1503],{"className":2449},[616],[573,2451],{"className":2452},[642,710],[573,2454],{"className":2455,"style":714},[642],[573,2457,2459,2462,2465,2474,2477,2480],{"className":2458},[607],[573,2460],{"className":2461,"style":721},[611],[573,2463,650],{"className":2464},[725],[573,2466,2468],{"className":2467},[616],[573,2469,2471],{"className":2470},[616],[573,2472,736],{"className":2473},[616,735],[573,2475],{"className":2476,"style":740},[642],[573,2478,2309],{"className":2479},[616],[573,2481,669],{"className":2482},[747]," gilt",[573,2485,2487],{"className":2486},[1875],[573,2488,2490,2559],{"className":2489},[576],[573,2491,2493],{"className":2492},[580],[582,2494,2495],{"xmlns":584,"display":1884},[586,2496,2497,2556],{},[589,2498,2499,2505,2507,2524,2526,2532,2534,2536,2538,2540,2542,2550,2552,2554],{},[970,2500,2501,2503],{},[1501,2502,2244],{},[1501,2504,2247],{},[635,2506,1499],{},[970,2508,2509,2521],{},[589,2510,2511,2513,2519],{},[635,2512,650],{"fence":603},[970,2514,2515,2517],{},[1501,2516,2244],{},[1501,2518,2343],{},[635,2520,669],{"fence":603},[1501,2522,2523],{},"25",[635,2525,637],{},[970,2527,2528,2530],{},[1501,2529,1503],{},[1501,2531,2523],{},[635,2533,1499],{},[1501,2535,1503],{},[642,2537],{},[642,2539],{"width":1917},[635,2541,650],{"stretchy":649},[589,2543,2544,2546,2548],{},[592,2545,594],{"mathvariant":655},[592,2547,658],{"mathvariant":655},[592,2549,661],{"mathvariant":655},[642,2551],{"width":664},[1501,2553,2309],{},[635,2555,669],{"stretchy":649},[596,2557,2558],{"encoding":598},"7^{100} = \\left(7^4\\right)^{25} \\equiv 1^{25} = 1 \\pmod{10}",[573,2560,2562,2611,2707,2754,2769],{"className":2561,"ariaHidden":603},[602],[573,2563,2565,2569,2602,2605,2608],{"className":2564},[607],[573,2566],{"className":2567,"style":2568},[611],"height:0.8641em;",[573,2570,2572,2575],{"className":2571},[616],[573,2573,2244],{"className":2574},[616],[573,2576,2578],{"className":2577},[1032],[573,2579,2581],{"className":2580},[1036],[573,2582,2584],{"className":2583},[1040],[573,2585,2587],{"className":2586,"style":2568},[1044],[573,2588,2590,2593],{"style":2589},"top:-3.113em;margin-right:0.05em;",[573,2591],{"className":2592,"style":1052},[1051],[573,2594,2596],{"className":2595},[1056,1057,1058,1059],[573,2597,2599],{"className":2598},[616,1059],[573,2600,2247],{"className":2601},[616,1059],[573,2603],{"className":2604,"style":689},[642],[573,2606,1499],{"className":2607},[693],[573,2609],{"className":2610,"style":689},[642],[573,2612,2614,2618,2698,2701,2704],{"className":2613},[607],[573,2615],{"className":2616,"style":2617},[611],"height:1.4181em;vertical-align:-0.35em;",[573,2619,2622,2670],{"className":2620},[2621],"minner",[573,2623,2625,2635,2664],{"className":2624},[2621],[573,2626,2630],{"className":2627,"style":2629},[725,2628],"delimcenter","top:0em;",[573,2631,650],{"className":2632},[2633,2634],"delimsizing","size1",[573,2636,2638,2641],{"className":2637},[616],[573,2639,2244],{"className":2640},[616],[573,2642,2644],{"className":2643},[1032],[573,2645,2647],{"className":2646},[1036],[573,2648,2650],{"className":2649},[1040],[573,2651,2653],{"className":2652,"style":2568},[1044],[573,2654,2655,2658],{"style":2589},[573,2656],{"className":2657,"style":1052},[1051],[573,2659,2661],{"className":2660},[1056,1057,1058,1059],[573,2662,2343],{"className":2663},[616,1059],[573,2665,2667],{"className":2666,"style":2629},[747,2628],[573,2668,669],{"className":2669},[2633,2634],[573,2671,2673],{"className":2672},[1032],[573,2674,2676],{"className":2675},[1036],[573,2677,2679],{"className":2678},[1040],[573,2680,2683],{"className":2681,"style":2682},[1044],"height:1.0681em;",[573,2684,2686,2689],{"style":2685},"top:-3.317em;margin-right:0.05em;",[573,2687],{"className":2688,"style":1052},[1051],[573,2690,2692],{"className":2691},[1056,1057,1058,1059],[573,2693,2695],{"className":2694},[616,1059],[573,2696,2523],{"className":2697},[616,1059],[573,2699],{"className":2700,"style":689},[642],[573,2702,637],{"className":2703},[693],[573,2705],{"className":2706,"style":689},[642],[573,2708,2710,2713,2745,2748,2751],{"className":2709},[607],[573,2711],{"className":2712,"style":2568},[611],[573,2714,2716,2719],{"className":2715},[616],[573,2717,1503],{"className":2718},[616],[573,2720,2722],{"className":2721},[1032],[573,2723,2725],{"className":2724},[1036],[573,2726,2728],{"className":2727},[1040],[573,2729,2731],{"className":2730,"style":2568},[1044],[573,2732,2733,2736],{"style":2589},[573,2734],{"className":2735,"style":1052},[1051],[573,2737,2739],{"className":2738},[1056,1057,1058,1059],[573,2740,2742],{"className":2741},[616,1059],[573,2743,2523],{"className":2744},[616,1059],[573,2746],{"className":2747,"style":689},[642],[573,2749,1499],{"className":2750},[693],[573,2752],{"className":2753,"style":689},[642],[573,2755,2757,2760,2763,2766],{"className":2756},[607],[573,2758],{"className":2759,"style":1558},[611],[573,2761,1503],{"className":2762},[616],[573,2764],{"className":2765},[642,710],[573,2767],{"className":2768,"style":2045},[642],[573,2770,2772,2775,2778,2787,2790,2793],{"className":2771},[607],[573,2773],{"className":2774,"style":721},[611],[573,2776,650],{"className":2777},[725],[573,2779,2781],{"className":2780},[616],[573,2782,2784],{"className":2783},[616],[573,2785,736],{"className":2786},[616,735],[573,2788],{"className":2789,"style":740},[642],[573,2791,2309],{"className":2792},[616],[573,2794,669],{"className":2795},[747],[565,2797,2798,2799,924],{},"Die letzte Ziffer ist ",[573,2800,2802,2815],{"className":2801},[576],[573,2803,2805],{"className":2804},[580],[582,2806,2807],{"xmlns":584},[586,2808,2809,2813],{},[589,2810,2811],{},[1501,2812,1503],{},[596,2814,1503],{"encoding":598},[573,2816,2818],{"className":2817,"ariaHidden":603},[602],[573,2819,2821,2824],{"className":2820},[607],[573,2822],{"className":2823,"style":1558},[611],[573,2825,1503],{"className":2826},[616],[560,2828,2830],{"id":2829},"typische-fehler","Typische Fehler",[2832,2833,2834,3442,4202],"ul",{},[2835,2836,2837,2840,2841,2992,2993,862,3021,3050,3051,3079,3080,3223,3224,3299,3300,3328,3329,3357,3358,924],"li",{},[569,2838,2839],{},"Unerlaubt kürzen."," ",[573,2842,2844,2893],{"className":2843},[576],[573,2845,2847],{"className":2846},[580],[582,2848,2849],{"xmlns":584},[586,2850,2851,2890],{},[589,2852,2853,2856,2858,2861,2863,2865,2867,2870,2872,2874,2876,2884,2886,2888],{},[1501,2854,2855],{},"2",[635,2857,1229],{},[1501,2859,2860],{},"3",[635,2862,637],{},[1501,2864,2855],{},[635,2866,1229],{},[1501,2868,2869],{},"8",[642,2871],{},[642,2873],{"width":646},[635,2875,650],{"stretchy":649},[589,2877,2878,2880,2882],{},[592,2879,594],{"mathvariant":655},[592,2881,658],{"mathvariant":655},[592,2883,661],{"mathvariant":655},[642,2885],{"width":664},[1501,2887,2309],{},[635,2889,669],{"stretchy":649},[596,2891,2892],{"encoding":598},"2 \\cdot 3 \\equiv 2 \\cdot 8 \\pmod{10}",[573,2894,2896,2914,2932,2950,2965],{"className":2895,"ariaHidden":603},[602],[573,2897,2899,2902,2905,2908,2911],{"className":2898},[607],[573,2900],{"className":2901,"style":1558},[611],[573,2903,2855],{"className":2904},[616],[573,2906],{"className":2907,"style":816},[642],[573,2909,1229],{"className":2910},[820],[573,2912],{"className":2913,"style":816},[642],[573,2915,2917,2920,2923,2926,2929],{"className":2916},[607],[573,2918],{"className":2919,"style":1558},[611],[573,2921,2860],{"className":2922},[616],[573,2924],{"className":2925,"style":689},[642],[573,2927,637],{"className":2928},[693],[573,2930],{"className":2931,"style":689},[642],[573,2933,2935,2938,2941,2944,2947],{"className":2934},[607],[573,2936],{"className":2937,"style":1558},[611],[573,2939,2855],{"className":2940},[616],[573,2942],{"className":2943,"style":816},[642],[573,2945,1229],{"className":2946},[820],[573,2948],{"className":2949,"style":816},[642],[573,2951,2953,2956,2959,2962],{"className":2952},[607],[573,2954],{"className":2955,"style":1558},[611],[573,2957,2869],{"className":2958},[616],[573,2960],{"className":2961},[642,710],[573,2963],{"className":2964,"style":714},[642],[573,2966,2968,2971,2974,2983,2986,2989],{"className":2967},[607],[573,2969],{"className":2970,"style":721},[611],[573,2972,650],{"className":2973},[725],[573,2975,2977],{"className":2976},[616],[573,2978,2980],{"className":2979},[616],[573,2981,736],{"className":2982},[616,735],[573,2984],{"className":2985,"style":740},[642],[573,2987,2309],{"className":2988},[616],[573,2990,669],{"className":2991},[747],", denn ",[573,2994,2996,3009],{"className":2995},[576],[573,2997,2999],{"className":2998},[580],[582,3000,3001],{"xmlns":584},[586,3002,3003,3007],{},[589,3004,3005],{},[1501,3006,1733],{},[596,3008,1733],{"encoding":598},[573,3010,3012],{"className":3011,"ariaHidden":603},[602],[573,3013,3015,3018],{"className":3014},[607],[573,3016],{"className":3017,"style":1558},[611],[573,3019,1733],{"className":3020},[616],[573,3022,3024,3038],{"className":3023},[576],[573,3025,3027],{"className":3026},[580],[582,3028,3029],{"xmlns":584},[586,3030,3031,3036],{},[589,3032,3033],{},[1501,3034,3035],{},"16",[596,3037,3035],{"encoding":598},[573,3039,3041],{"className":3040,"ariaHidden":603},[602],[573,3042,3044,3047],{"className":3043},[607],[573,3045],{"className":3046,"style":1558},[611],[573,3048,3035],{"className":3049},[616]," haben beide den Rest ",[573,3052,3054,3067],{"className":3053},[576],[573,3055,3057],{"className":3056},[580],[582,3058,3059],{"xmlns":584},[586,3060,3061,3065],{},[589,3062,3063],{},[1501,3064,1733],{},[596,3066,1733],{"encoding":598},[573,3068,3070],{"className":3069,"ariaHidden":603},[602],[573,3071,3073,3076],{"className":3072},[607],[573,3074],{"className":3075,"style":1558},[611],[573,3077,1733],{"className":3078},[616],". Trotzdem ist ",[573,3081,3083,3122],{"className":3082},[576],[573,3084,3086],{"className":3085},[580],[582,3087,3088],{"xmlns":584},[586,3089,3090,3119],{},[589,3091,3092,3094,3097,3099,3101,3103,3105,3113,3115,3117],{},[1501,3093,2860],{},[635,3095,3096],{},"≢",[1501,3098,2869],{},[642,3100],{},[642,3102],{"width":646},[635,3104,650],{"stretchy":649},[589,3106,3107,3109,3111],{},[592,3108,594],{"mathvariant":655},[592,3110,658],{"mathvariant":655},[592,3112,661],{"mathvariant":655},[642,3114],{"width":664},[1501,3116,2309],{},[635,3118,669],{"stretchy":649},[596,3120,3121],{"encoding":598},"3 \\not\\equiv 8 \\pmod{10}",[573,3123,3125,3181,3196],{"className":3124,"ariaHidden":603},[602],[573,3126,3128,3132,3135,3138,3171,3175,3178],{"className":3127},[607],[573,3129],{"className":3130,"style":3131},[611],"height:0.8889em;vertical-align:-0.1944em;",[573,3133,2860],{"className":3134},[616],[573,3136],{"className":3137,"style":689},[642],[573,3139,3141],{"className":3140},[693],[573,3142,3145],{"className":3143},[616,3144],"vbox",[573,3146,3149],{"className":3147},[3148],"thinbox",[573,3150,3153,3156,3167],{"className":3151},[3152],"rlap",[573,3154],{"className":3155,"style":3131},[611],[573,3157,3160],{"className":3158},[3159],"inner",[573,3161,3163],{"className":3162},[616],[573,3164,3166],{"className":3165},[693],"",[573,3168],{"className":3169},[3170],"fix",[573,3172],{"className":3173},[642,3174],"nobreak",[573,3176,637],{"className":3177},[693],[573,3179],{"className":3180,"style":689},[642],[573,3182,3184,3187,3190,3193],{"className":3183},[607],[573,3185],{"className":3186,"style":1558},[611],[573,3188,2869],{"className":3189},[616],[573,3191],{"className":3192},[642,710],[573,3194],{"className":3195,"style":714},[642],[573,3197,3199,3202,3205,3214,3217,3220],{"className":3198},[607],[573,3200],{"className":3201,"style":721},[611],[573,3203,650],{"className":3204},[725],[573,3206,3208],{"className":3207},[616],[573,3209,3211],{"className":3210},[616],[573,3212,736],{"className":3213},[616,735],[573,3215],{"className":3216,"style":740},[642],[573,3218,2309],{"className":3219},[616],[573,3221,669],{"className":3222},[747],", weil ",[573,3225,3227,3250],{"className":3226},[576],[573,3228,3230],{"className":3229},[580],[582,3231,3232],{"xmlns":584},[586,3233,3234,3247],{},[589,3235,3236,3238,3240,3242,3244],{},[1501,3237,2869],{},[635,3239,794],{},[1501,3241,2860],{},[635,3243,1499],{},[1501,3245,3246],{},"5",[596,3248,3249],{"encoding":598},"8 - 3 = 5",[573,3251,3253,3272,3290],{"className":3252,"ariaHidden":603},[602],[573,3254,3256,3260,3263,3266,3269],{"className":3255},[607],[573,3257],{"className":3258,"style":3259},[611],"height:0.7278em;vertical-align:-0.0833em;",[573,3261,2869],{"className":3262},[616],[573,3264],{"className":3265,"style":816},[642],[573,3267,794],{"className":3268},[820],[573,3270],{"className":3271,"style":816},[642],[573,3273,3275,3278,3281,3284,3287],{"className":3274},[607],[573,3276],{"className":3277,"style":1558},[611],[573,3279,2860],{"className":3280},[616],[573,3282],{"className":3283,"style":689},[642],[573,3285,1499],{"className":3286},[693],[573,3288],{"className":3289,"style":689},[642],[573,3291,3293,3296],{"className":3292},[607],[573,3294],{"className":3295,"style":1558},[611],[573,3297,3246],{"className":3298},[616]," nicht durch ",[573,3301,3303,3316],{"className":3302},[576],[573,3304,3306],{"className":3305},[580],[582,3307,3308],{"xmlns":584},[586,3309,3310,3314],{},[589,3311,3312],{},[1501,3313,2309],{},[596,3315,2309],{"encoding":598},[573,3317,3319],{"className":3318,"ariaHidden":603},[602],[573,3320,3322,3325],{"className":3321},[607],[573,3323],{"className":3324,"style":1558},[611],[573,3326,2309],{"className":3327},[616]," teilbar ist. Kürzen durch ",[573,3330,3332,3345],{"className":3331},[576],[573,3333,3335],{"className":3334},[580],[582,3336,3337],{"xmlns":584},[586,3338,3339,3343],{},[589,3340,3341],{},[1501,3342,2855],{},[596,3344,2855],{"encoding":598},[573,3346,3348],{"className":3347,"ariaHidden":603},[602],[573,3349,3351,3354],{"className":3350},[607],[573,3352],{"className":3353,"style":1558},[611],[573,3355,2855],{"className":3356},[616]," ist hier verboten, da ",[573,3359,3361,3391],{"className":3360},[576],[573,3362,3364],{"className":3363},[580],[582,3365,3366],{"xmlns":584},[586,3367,3368,3388],{},[589,3369,3370,3372,3374,3376,3378,3380,3382,3384,3386],{},[592,3371,1482],{"mathvariant":655},[635,3373,1485],{},[635,3375,650],{"stretchy":649},[1501,3377,2855],{},[635,3379,1492],{"separator":603},[1501,3381,2309],{},[635,3383,669],{"stretchy":649},[635,3385,1499],{},[1501,3387,2855],{},[596,3389,3390],{"encoding":598},"\\operatorname{ggT}(2, 10) = 2",[573,3392,3394,3433],{"className":3393,"ariaHidden":603},[602],[573,3395,3397,3400,3406,3409,3412,3415,3418,3421,3424,3427,3430],{"className":3396},[607],[573,3398],{"className":3399,"style":721},[611],[573,3401,3403],{"className":3402},[1519],[573,3404,1482],{"className":3405},[616,735],[573,3407,650],{"className":3408},[725],[573,3410,2855],{"className":3411},[616],[573,3413,1492],{"className":3414},[1532],[573,3416],{"className":3417,"style":1536},[642],[573,3419,2309],{"className":3420},[616],[573,3422,669],{"className":3423},[747],[573,3425],{"className":3426,"style":689},[642],[573,3428,1499],{"className":3429},[693],[573,3431],{"className":3432,"style":689},[642],[573,3434,3436,3439],{"className":3435},[607],[573,3437],{"className":3438,"style":1558},[611],[573,3440,2855],{"className":3441},[616],[2835,3443,3444,3476,3477,3648,3649,3749,3750,3831,3832,3877,3878,3962,3963,862,4029,924],{},[569,3445,3446,3447,3475],{},"Im Exponenten modulo ",[573,3448,3450,3463],{"className":3449},[576],[573,3451,3453],{"className":3452},[580],[582,3454,3455],{"xmlns":584},[586,3456,3457,3461],{},[589,3458,3459],{},[592,3460,594],{},[596,3462,594],{"encoding":598},[573,3464,3466],{"className":3465,"ariaHidden":603},[602],[573,3467,3469,3472],{"className":3468},[607],[573,3470],{"className":3471,"style":612},[611],[573,3473,594],{"className":3474},[616,617]," reduzieren."," Es ist ",[573,3478,3480,3532],{"className":3479},[576],[573,3481,3483],{"className":3482},[580],[582,3484,3485],{"xmlns":584},[586,3486,3487,3529],{},[589,3488,3489,3495,3497,3500,3502,3505,3507,3509,3511,3513,3515,3523,3525,3527],{},[970,3490,3491,3493],{},[1501,3492,2860],{},[1501,3494,2309],{},[635,3496,1499],{},[1501,3498,3499],{},"59",[2100,3501,2102],{},[1501,3503,3504],{},"049",[635,3506,637],{},[1501,3508,1699],{},[642,3510],{},[642,3512],{"width":646},[635,3514,650],{"stretchy":649},[589,3516,3517,3519,3521],{},[592,3518,594],{"mathvariant":655},[592,3520,658],{"mathvariant":655},[592,3522,661],{"mathvariant":655},[642,3524],{"width":664},[1501,3526,2309],{},[635,3528,669],{"stretchy":649},[596,3530,3531],{"encoding":598},"3^{10} = 59\\,049 \\equiv 9 \\pmod{10}",[573,3533,3535,3582,3606,3621],{"className":3534,"ariaHidden":603},[602],[573,3536,3538,3541,3573,3576,3579],{"className":3537},[607],[573,3539],{"className":3540,"style":2260},[611],[573,3542,3544,3547],{"className":3543},[616],[573,3545,2860],{"className":3546},[616],[573,3548,3550],{"className":3549},[1032],[573,3551,3553],{"className":3552},[1036],[573,3554,3556],{"className":3555},[1040],[573,3557,3559],{"className":3558,"style":2260},[1044],[573,3560,3561,3564],{"style":1047},[573,3562],{"className":3563,"style":1052},[1051],[573,3565,3567],{"className":3566},[1056,1057,1058,1059],[573,3568,3570],{"className":3569},[616,1059],[573,3571,2309],{"className":3572},[616,1059],[573,3574],{"className":3575,"style":689},[642],[573,3577,1499],{"className":3578},[693],[573,3580],{"className":3581,"style":689},[642],[573,3583,3585,3588,3591,3594,3597,3600,3603],{"className":3584},[607],[573,3586],{"className":3587,"style":1558},[611],[573,3589,3499],{"className":3590},[616],[573,3592],{"className":3593,"style":1536},[642],[573,3595,3504],{"className":3596},[616],[573,3598],{"className":3599,"style":689},[642],[573,3601,637],{"className":3602},[693],[573,3604],{"className":3605,"style":689},[642],[573,3607,3609,3612,3615,3618],{"className":3608},[607],[573,3610],{"className":3611,"style":1558},[611],[573,3613,1699],{"className":3614},[616],[573,3616],{"className":3617},[642,710],[573,3619],{"className":3620,"style":714},[642],[573,3622,3624,3627,3630,3639,3642,3645],{"className":3623},[607],[573,3625],{"className":3626,"style":721},[611],[573,3628,650],{"className":3629},[725],[573,3631,3633],{"className":3632},[616],[573,3634,3636],{"className":3635},[616],[573,3637,736],{"className":3638},[616,735],[573,3640],{"className":3641,"style":740},[642],[573,3643,2309],{"className":3644},[616],[573,3646,669],{"className":3647},[747],". Wer den Exponenten durch ",[573,3650,3652,3688],{"className":3651},[576],[573,3653,3655],{"className":3654},[580],[582,3656,3657],{"xmlns":584},[586,3658,3659,3685],{},[589,3660,3661,3663,3666,3677,3679,3681,3683],{},[1501,3662,2309],{},[2100,3664,3665],{}," ",[635,3667,3669],{"lspace":3668,"rspace":3668},"0.22em",[589,3670,3671,3673,3675],{},[592,3672,594],{"mathvariant":655},[592,3674,658],{"mathvariant":655},[592,3676,661],{"mathvariant":655},[2100,3678,3665],{},[1501,3680,2309],{},[635,3682,1499],{},[1501,3684,1912],{},[596,3686,3687],{"encoding":598},"10 \\bmod 10 = 0",[573,3689,3691,3722,3740],{"className":3690,"ariaHidden":603},[602],[573,3692,3694,3697,3700,3704,3707,3716,3719],{"className":3693},[607],[573,3695],{"className":3696,"style":703},[611],[573,3698,2309],{"className":3699},[616],[573,3701],{"className":3702,"style":3703},[642],"margin-right:0.0556em;",[573,3705],{"className":3706,"style":816},[642],[573,3708,3710],{"className":3709},[820],[573,3711,3713],{"className":3712},[616],[573,3714,736],{"className":3715},[616,735],[573,3717],{"className":3718,"style":3703},[642],[573,3720],{"className":3721,"style":816},[642],[573,3723,3725,3728,3731,3734,3737],{"className":3724},[607],[573,3726],{"className":3727,"style":1558},[611],[573,3729,2309],{"className":3730},[616],[573,3732],{"className":3733,"style":689},[642],[573,3735,1499],{"className":3736},[693],[573,3738],{"className":3739,"style":689},[642],[573,3741,3743,3746],{"className":3742},[607],[573,3744],{"className":3745,"style":1558},[611],[573,3747,1912],{"className":3748},[616]," ersetzt, erhält fälschlich ",[573,3751,3753,3775],{"className":3752},[576],[573,3754,3756],{"className":3755},[580],[582,3757,3758],{"xmlns":584},[586,3759,3760,3772],{},[589,3761,3762,3768,3770],{},[970,3763,3764,3766],{},[1501,3765,2860],{},[1501,3767,1912],{},[635,3769,1499],{},[1501,3771,1503],{},[596,3773,3774],{"encoding":598},"3^0 = 1",[573,3776,3778,3822],{"className":3777,"ariaHidden":603},[602],[573,3779,3781,3784,3813,3816,3819],{"className":3780},[607],[573,3782],{"className":3783,"style":2260},[611],[573,3785,3787,3790],{"className":3786},[616],[573,3788,2860],{"className":3789},[616],[573,3791,3793],{"className":3792},[1032],[573,3794,3796],{"className":3795},[1036],[573,3797,3799],{"className":3798},[1040],[573,3800,3802],{"className":3801,"style":2260},[1044],[573,3803,3804,3807],{"style":1047},[573,3805],{"className":3806,"style":1052},[1051],[573,3808,3810],{"className":3809},[1056,1057,1058,1059],[573,3811,1912],{"className":3812},[616,1059],[573,3814],{"className":3815,"style":689},[642],[573,3817,1499],{"className":3818},[693],[573,3820],{"className":3821,"style":689},[642],[573,3823,3825,3828],{"className":3824},[607],[573,3826],{"className":3827,"style":1558},[611],[573,3829,1503],{"className":3830},[616],". Exponenten darf man nur modulo einer Periode reduzieren, etwa modulo ",[573,3833,3835,3856],{"className":3834},[576],[573,3836,3838],{"className":3837},[580],[582,3839,3840],{"xmlns":584},[586,3841,3842,3853],{},[589,3843,3844,3847,3849,3851],{},[592,3845,3846],{},"φ",[635,3848,650],{"stretchy":649},[592,3850,594],{},[635,3852,669],{"stretchy":649},[596,3854,3855],{"encoding":598},"\\varphi(m)",[573,3857,3859],{"className":3858,"ariaHidden":603},[602],[573,3860,3862,3865,3868,3871,3874],{"className":3861},[607],[573,3863],{"className":3864,"style":721},[611],[573,3866,3846],{"className":3867},[616,617],[573,3869,650],{"className":3870},[725],[573,3872,594],{"className":3873},[616,617],[573,3875,669],{"className":3876},[747],", falls ",[573,3879,3881,3911],{"className":3880},[576],[573,3882,3884],{"className":3883},[580],[582,3885,3886],{"xmlns":584},[586,3887,3888,3908],{},[589,3889,3890,3892,3894,3896,3898,3900,3902,3904,3906],{},[592,3891,1482],{"mathvariant":655},[635,3893,1485],{},[635,3895,650],{"stretchy":649},[592,3897,633],{},[635,3899,1492],{"separator":603},[592,3901,594],{},[635,3903,669],{"stretchy":649},[635,3905,1499],{},[1501,3907,1503],{},[596,3909,3910],{"encoding":598},"\\operatorname{ggT}(a, m) = 1",[573,3912,3914,3953],{"className":3913,"ariaHidden":603},[602],[573,3915,3917,3920,3926,3929,3932,3935,3938,3941,3944,3947,3950],{"className":3916},[607],[573,3918],{"className":3919,"style":721},[611],[573,3921,3923],{"className":3922},[1519],[573,3924,1482],{"className":3925},[616,735],[573,3927,650],{"className":3928},[725],[573,3930,633],{"className":3931},[616,617],[573,3933,1492],{"className":3934},[1532],[573,3936],{"className":3937,"style":1536},[642],[573,3939,594],{"className":3940},[616,617],[573,3942,669],{"className":3943},[747],[573,3945],{"className":3946,"style":689},[642],[573,3948,1499],{"className":3949},[693],[573,3951],{"className":3952,"style":689},[642],[573,3954,3956,3959],{"className":3955},[607],[573,3957],{"className":3958,"style":1558},[611],[573,3960,1503],{"className":3961},[616]," (Satz von Euler): ",[573,3964,3966,3990],{"className":3965},[576],[573,3967,3969],{"className":3968},[580],[582,3970,3971],{"xmlns":584},[586,3972,3973,3987],{},[589,3974,3975,3977,3979,3981,3983,3985],{},[592,3976,3846],{},[635,3978,650],{"stretchy":649},[1501,3980,2309],{},[635,3982,669],{"stretchy":649},[635,3984,1499],{},[1501,3986,2343],{},[596,3988,3989],{"encoding":598},"\\varphi(10) = 4",[573,3991,3993,4020],{"className":3992,"ariaHidden":603},[602],[573,3994,3996,3999,4002,4005,4008,4011,4014,4017],{"className":3995},[607],[573,3997],{"className":3998,"style":721},[611],[573,4000,3846],{"className":4001},[616,617],[573,4003,650],{"className":4004},[725],[573,4006,2309],{"className":4007},[616],[573,4009,669],{"className":4010},[747],[573,4012],{"className":4013,"style":689},[642],[573,4015,1499],{"className":4016},[693],[573,4018],{"className":4019,"style":689},[642],[573,4021,4023,4026],{"className":4022},[607],[573,4024],{"className":4025,"style":1558},[611],[573,4027,2343],{"className":4028},[616],[573,4030,4032,4080],{"className":4031},[576],[573,4033,4035],{"className":4034},[580],[582,4036,4037],{"xmlns":584},[586,4038,4039,4077],{},[589,4040,4041,4065,4067,4073,4075],{},[970,4042,4043,4045],{},[1501,4044,2860],{},[589,4046,4047,4049,4051,4061,4063],{},[1501,4048,2309],{},[2100,4050,3665],{},[635,4052,4053],{"lspace":3668,"rspace":3668},[589,4054,4055,4057,4059],{},[592,4056,594],{"mathvariant":655},[592,4058,658],{"mathvariant":655},[592,4060,661],{"mathvariant":655},[2100,4062,3665],{},[1501,4064,2343],{},[635,4066,1499],{},[970,4068,4069,4071],{},[1501,4070,2860],{},[1501,4072,2855],{},[635,4074,1499],{},[1501,4076,1699],{},[596,4078,4079],{"encoding":598},"3^{10 \\bmod 4} = 3^2 = 9",[573,4081,4083,4149,4193],{"className":4082,"ariaHidden":603},[602],[573,4084,4086,4089,4140,4143,4146],{"className":4085},[607],[573,4087],{"className":4088,"style":1022},[611],[573,4090,4092,4095],{"className":4091},[616],[573,4093,2860],{"className":4094},[616],[573,4096,4098],{"className":4097},[1032],[573,4099,4101],{"className":4100},[1036],[573,4102,4104],{"className":4103},[1040],[573,4105,4107],{"className":4106,"style":1022},[1044],[573,4108,4109,4112],{"style":1047},[573,4110],{"className":4111,"style":1052},[1051],[573,4113,4115],{"className":4114},[1056,1057,1058,1059],[573,4116,4118,4121,4125,4134,4137],{"className":4117},[616,1059],[573,4119,2309],{"className":4120},[616,1059],[573,4122],{"className":4123,"style":4124},[642,1059],"margin-right:0.3253em;",[573,4126,4128],{"className":4127},[820,1059],[573,4129,4131],{"className":4130},[616,1059],[573,4132,736],{"className":4133},[616,735,1059],[573,4135],{"className":4136,"style":4124},[642,1059],[573,4138,2343],{"className":4139},[616,1059],[573,4141],{"className":4142,"style":689},[642],[573,4144,1499],{"className":4145},[693],[573,4147],{"className":4148,"style":689},[642],[573,4150,4152,4155,4184,4187,4190],{"className":4151},[607],[573,4153],{"className":4154,"style":2260},[611],[573,4156,4158,4161],{"className":4157},[616],[573,4159,2860],{"className":4160},[616],[573,4162,4164],{"className":4163},[1032],[573,4165,4167],{"className":4166},[1036],[573,4168,4170],{"className":4169},[1040],[573,4171,4173],{"className":4172,"style":2260},[1044],[573,4174,4175,4178],{"style":1047},[573,4176],{"className":4177,"style":1052},[1051],[573,4179,4181],{"className":4180},[1056,1057,1058,1059],[573,4182,2855],{"className":4183},[616,1059],[573,4185],{"className":4186,"style":689},[642],[573,4188,1499],{"className":4189},[693],[573,4191],{"className":4192,"style":689},[642],[573,4194,4196,4199],{"className":4195},[607],[573,4197],{"className":4198,"style":1558},[611],[573,4200,1699],{"className":4201},[616],[2835,4203,4204,2840,4207,4316,4317,4345,4346,862,4374,924],{},[569,4205,4206],{},"Negative Zahlen falsch reduzieren.",[573,4208,4210,4250],{"className":4209},[576],[573,4211,4213],{"className":4212},[580],[582,4214,4215],{"xmlns":584},[586,4216,4217,4247],{},[589,4218,4219,4221,4223,4225,4227,4229,4231,4233,4241,4243,4245],{},[635,4220,794],{},[1501,4222,2860],{},[635,4224,637],{},[1501,4226,2244],{},[642,4228],{},[642,4230],{"width":646},[635,4232,650],{"stretchy":649},[589,4234,4235,4237,4239],{},[592,4236,594],{"mathvariant":655},[592,4238,658],{"mathvariant":655},[592,4240,661],{"mathvariant":655},[642,4242],{"width":664},[1501,4244,2309],{},[635,4246,669],{"stretchy":649},[596,4248,4249],{"encoding":598},"-3 \\equiv 7 \\pmod{10}",[573,4251,4253,4274,4289],{"className":4252,"ariaHidden":603},[602],[573,4254,4256,4259,4262,4265,4268,4271],{"className":4255},[607],[573,4257],{"className":4258,"style":3259},[611],[573,4260,794],{"className":4261},[616],[573,4263,2860],{"className":4264},[616],[573,4266],{"className":4267,"style":689},[642],[573,4269,637],{"className":4270},[693],[573,4272],{"className":4273,"style":689},[642],[573,4275,4277,4280,4283,4286],{"className":4276},[607],[573,4278],{"className":4279,"style":1558},[611],[573,4281,2244],{"className":4282},[616],[573,4284],{"className":4285},[642,710],[573,4287],{"className":4288,"style":714},[642],[573,4290,4292,4295,4298,4307,4310,4313],{"className":4291},[607],[573,4293],{"className":4294,"style":721},[611],[573,4296,650],{"className":4297},[725],[573,4299,4301],{"className":4300},[616],[573,4302,4304],{"className":4303},[616],[573,4305,736],{"className":4306},[616,735],[573,4308],{"className":4309,"style":740},[642],[573,4311,2309],{"className":4312},[616],[573,4314,669],{"className":4315},[747],", nicht ",[573,4318,4320,4333],{"className":4319},[576],[573,4321,4323],{"className":4322},[580],[582,4324,4325],{"xmlns":584},[586,4326,4327,4331],{},[589,4328,4329],{},[1501,4330,2860],{},[596,4332,2860],{"encoding":598},[573,4334,4336],{"className":4335,"ariaHidden":603},[602],[573,4337,4339,4342],{"className":4338},[607],[573,4340],{"className":4341,"style":1558},[611],[573,4343,2860],{"className":4344},[616],". Der Rest liegt immer zwischen ",[573,4347,4349,4362],{"className":4348},[576],[573,4350,4352],{"className":4351},[580],[582,4353,4354],{"xmlns":584},[586,4355,4356,4360],{},[589,4357,4358],{},[1501,4359,1912],{},[596,4361,1912],{"encoding":598},[573,4363,4365],{"className":4364,"ariaHidden":603},[602],[573,4366,4368,4371],{"className":4367},[607],[573,4369],{"className":4370,"style":1558},[611],[573,4372,1912],{"className":4373},[616],[573,4375,4377,4395],{"className":4376},[576],[573,4378,4380],{"className":4379},[580],[582,4381,4382],{"xmlns":584},[586,4383,4384,4392],{},[589,4385,4386,4388,4390],{},[592,4387,594],{},[635,4389,794],{},[1501,4391,1503],{},[596,4393,4394],{"encoding":598},"m - 1",[573,4396,4398,4416],{"className":4397,"ariaHidden":603},[602],[573,4399,4401,4404,4407,4410,4413],{"className":4400},[607],[573,4402],{"className":4403,"style":809},[611],[573,4405,594],{"className":4406},[616,617],[573,4408],{"className":4409,"style":816},[642],[573,4411,794],{"className":4412},[820],[573,4414],{"className":4415,"style":816},[642],[573,4417,4419,4422],{"className":4418},[607],[573,4420],{"className":4421,"style":1558},[611],[573,4423,1503],{"className":4424},[616],{"title":4426,"searchDepth":4427,"depth":4427,"links":4428},"",2,[4429,4430,4431],{"id":562,"depth":4427,"text":563},{"id":1623,"depth":4427,"text":1624},{"id":2829,"depth":4427,"text":2830},[4433,4440,4444,4446,4448,4450,4453,4455,4457,4459,4461,4464,4466,4469,4471,4473],{"lhs":4434,"rhs":4435,"where":4436},"mod(a*b, m)","mod(mod(a, m)*mod(b, m), m)",{"a":4437,"b":4438,"m":4439},123,456,9,{"lhs":4441,"rhs":4442,"where":4443},"mod(a + b, m)","mod(mod(a, m) + mod(b, m), m)",{"a":4437,"b":4438,"m":4439},{"lhs":4445,"rhs":1733},"mod(123, 9)",{"lhs":4447,"rhs":1733},"mod(456, 9)",{"lhs":4449,"rhs":1912},"mod(6*6, 9)",{"lhs":4451,"rhs":4452},"123*456","56088",{"lhs":4454,"rhs":1912},"mod(56088, 9)",{"lhs":4456,"rhs":2348},"7^4",{"lhs":4458,"rhs":1503},"mod(7^4, 10)",{"lhs":4460,"rhs":2247},"4*25",{"lhs":4462,"rhs":4463},"mod(2*3, 10)","mod(2*8, 10)",{"lhs":4465,"rhs":3246},"mod(8 - 3, 10)",{"lhs":4467,"rhs":4468},"3^10","59049",{"lhs":4470,"rhs":1699},"mod(3^10, 10)",{"lhs":4472,"rhs":1699},"mod(3^mod(10, 4), 10)",{"lhs":4474,"rhs":2244},"mod(-3, 10)","a, a', b, b' \\in \\mathbb{Z},\\ m \\in \\mathbb{N},\\ m \\geq 2","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">Z\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, a', b, b' \\in \\mathbb{Z},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">Z\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>m\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">N\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">m \\in \\mathbb{N},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">N\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>m\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">m \\geq 2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7719em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","modular-arithmetic","a \\equiv a',\\ b \\equiv b' \\!\\!\\pmod m \\;\\Rightarrow\\; a \\pm b \\equiv a' \\pm b',\\ \\ a \\cdot b \\equiv a' \\cdot b' \\!\\!\\pmod m","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>≡\u003C\u002Fmo>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>≡\u003C\u002Fmo>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmspace>\u003C\u002Fmspace>\u003Cmspace width=\"1em\"\u002F>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmrow>\u003Cmi mathvariant=\"normal\">m\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">o\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">d\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmspace width=\"0.3333em\"\u002F>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>±\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>≡\u003C\u002Fmo>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>±\u003C\u002Fmo>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo>≡\u003C\u002Fmo>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmspace>\u003C\u002Fmspace>\u003Cmspace width=\"1em\"\u002F>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmrow>\u003Cmi mathvariant=\"normal\">m\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">o\u003C\u002Fmi>\u003Cmi 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, b\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4507,"meaning":4508,"symbolHtml":4509},"a', b'","ganze Zahlen mit demselben Rest modulo m wie a bzw. b","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a', b'\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord 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mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":672,"meaning":4514,"symbolHtml":4515},"a ist kongruent zu b modulo m, das heißt m teilt a − b","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>≡\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmspace>\u003C\u002Fmspace>\u003Cmspace width=\"0.4444em\"\u002F>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmrow>\u003Cmi mathvariant=\"normal\">m\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">o\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">d\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmspace width=\"0.3333em\"\u002F>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a \\equiv b \\pmod m\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4637em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≡\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mspace allowbreak\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.4444em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">mod\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.3333em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","KPoq73chNDkImXGn2nP6jmBFiJ9GyHYKApkh21Kh0Qw",[4518,4521,4524,4527],{"stem":4519,"title":4520,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fteilbarkeitsregeln","Teilbarkeitsregeln",{"stem":4522,"title":4523,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fchinesischer-restsatz","Chinesischer Restsatz",{"stem":4525,"title":4526,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Feuklidischer-algorithmus","Euklidischer Algorithmus",{"stem":4528,"title":4529,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fkleiner-satz-von-fermat","Kleiner Satz von Fermat",1793824276779]