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p^{k-1}",[573,1102,1104,1171,1216],{"className":1103,"ariaHidden":598},[622],[573,1105,1107,1111,1114,1117,1159,1162,1165,1168],{"className":1106},[626],[573,1108],{"className":1109,"style":1110},[630],"height:1.0991em;vertical-align:-0.25em;",[573,1112,848],{"className":1113},[635,670],[573,1115,852],{"className":1116},[887],[573,1118,1120,1123],{"className":1119},[635],[573,1121,565],{"className":1122},[635,670],[573,1124,1127],{"className":1125},[1126],"msupsub",[573,1128,1131],{"className":1129},[1130],"vlist-t",[573,1132,1135],{"className":1133},[1134],"vlist-r",[573,1136,1140],{"className":1137,"style":1139},[1138],"vlist","height:0.8491em;",[573,1141,1143,1148],{"style":1142},"top:-3.063em;margin-right:0.05em;",[573,1144],{"className":1145,"style":1147},[1146],"pstrut","height:2.7em;",[573,1149,1155],{"className":1150},[1151,1152,1153,1154],"sizing","reset-size6","size3","mtight",[573,1156,1073],{"className":1157,"style":1158},[635,670,1154],"margin-right:0.0315em;",[573,1160,857],{"className":1161},[894],[573,1163],{"className":1164,"style":898},[643],[573,1166,860],{"className":1167},[902],[573,1169],{"className":1170,"style":898},[643],[573,1172,1174,1178,1207,1210,1213],{"className":1173},[626],[573,1175],{"className":1176,"style":1177},[630],"height:1.0435em;vertical-align:-0.1944em;",[573,1179,1181,1184],{"className":1180},[635],[573,1182,565],{"className":1183},[635,670],[573,1185,1187],{"className":1186},[1126],[573,1188,1190],{"className":1189},[1130],[573,1191,1193],{"className":1192},[1134],[573,1194,1196],{"className":1195,"style":1139},[1138],[573,1197,1198,1201],{"style":1142},[573,1199],{"className":1200,"style":1147},[1146],[573,1202,1204],{"className":1203},[1151,1152,1153,1154],[573,1205,1073],{"className":1206,"style":1158},[635,670,1154],[573,1208],{"className":1209,"style":919},[643],[573,1211,865],{"className":1212},[923],[573,1214],{"className":1215,"style":919},[643],[573,1217,1219,1222],{"className":1218},[626],[573,1220],{"className":1221,"style":1177},[630],[573,1223,1225,1228],{"className":1224},[635],[573,1226,565],{"className":1227},[635,670],[573,1229,1231],{"className":1230},[1126],[573,1232,1234],{"className":1233},[1130],[573,1235,1237],{"className":1236},[1134],[573,1238,1240],{"className":1239,"style":1139},[1138],[573,1241,1242,1245],{"style":1142},[573,1243],{"className":1244,"style":1147},[1146],[573,1246,1248],{"className":1247},[1151,1152,1153,1154],[573,1249,1251,1254,1257],{"className":1250},[635,1154],[573,1252,1073],{"className":1253,"style":1158},[635,670,1154],[573,1255,865],{"className":1256},[923,1154],[573,1258,594],{"className":1259},[635,1154],", denn genau die ",[573,1262,1264,1288],{"className":1263},[576],[573,1265,1267],{"className":1266},[580],[582,1268,1269],{"xmlns":584},[586,1270,1271,1285],{},[589,1272,1273],{},[1067,1274,1275,1277],{},[611,1276,565],{},[589,1278,1279,1281,1283],{},[611,1280,1073],{},[596,1282,865],{},[592,1284,594],{},[615,1286,1287],{"encoding":617},"p^{k-1}",[573,1289,1291],{"className":1290,"ariaHidden":598},[622],[573,1292,1294,1297],{"className":1293},[626],[573,1295],{"className":1296,"style":1177},[630],[573,1298,1300,1303],{"className":1299},[635],[573,1301,565],{"className":1302},[635,670],[573,1304,1306],{"className":1305},[1126],[573,1307,1309],{"className":1308},[1130],[573,1310,1312],{"className":1311},[1134],[573,1313,1315],{"className":1314,"style":1139},[1138],[573,1316,1317,1320],{"style":1142},[573,1318],{"className":1319,"style":1147},[1146],[573,1321,1323],{"className":1322},[1151,1152,1153,1154],[573,1324,1326,1329,1332],{"className":1325},[635,1154],[573,1327,1073],{"className":1328,"style":1158},[635,670,1154],[573,1330,865],{"className":1331},[923,1154],[573,1333,594],{"className":1334},[635,1154]," Vielfachen von ",[573,1337,1339,1352],{"className":1338},[576],[573,1340,1342],{"className":1341},[580],[582,1343,1344],{"xmlns":584},[586,1345,1346,1350],{},[589,1347,1348],{},[611,1349,565],{},[615,1351,565],{"encoding":617},[573,1353,1355],{"className":1354,"ariaHidden":598},[622],[573,1356,1358,1361],{"className":1357},[626],[573,1359],{"className":1360,"style":829},[630],[573,1362,565],{"className":1363},[635,670]," fallen weg.",[801,1366,1367,1370,1371,1414,1415,1046],{},[569,1368,1369],{},"Multiplikativität:"," Für teilerfremde ",[573,1372,1374,1393],{"className":1373},[576],[573,1375,1377],{"className":1376},[580],[582,1378,1379],{"xmlns":584},[586,1380,1381,1390],{},[589,1382,1383,1386,1388],{},[611,1384,1385],{},"m",[596,1387,599],{"separator":598},[611,1389,613],{},[615,1391,1392],{"encoding":617},"m, n",[573,1394,1396],{"className":1395,"ariaHidden":598},[622],[573,1397,1399,1402,1405,1408,1411],{"className":1398},[626],[573,1400],{"className":1401,"style":829},[630],[573,1403,1385],{"className":1404},[635,670],[573,1406,599],{"className":1407},[639],[573,1409],{"className":1410,"style":644},[643],[573,1412,613],{"className":1413},[635,670]," gilt ",[573,1416,1418,1463],{"className":1417},[576],[573,1419,1421],{"className":1420},[580],[582,1422,1423],{"xmlns":584},[586,1424,1425,1460],{},[589,1426,1427,1429,1431,1433,1436,1438,1440,1442,1444,1446,1448,1450,1452,1454,1456,1458],{},[611,1428,848],{},[596,1430,852],{"stretchy":851},[611,1432,1385],{},[596,1434,1435],{},"⋅",[611,1437,613],{},[596,1439,857],{"stretchy":851},[596,1441,860],{},[611,1443,848],{},[596,1445,852],{"stretchy":851},[611,1447,1385],{},[596,1449,857],{"stretchy":851},[596,1451,1435],{},[611,1453,848],{},[596,1455,852],{"stretchy":851},[611,1457,613],{},[596,1459,857],{"stretchy":851},[615,1461,1462],{"encoding":617},"\\varphi(m \\cdot n) = \\varphi(m) \\cdot \\varphi(n)",[573,1464,1466,1490,1511,1538],{"className":1465,"ariaHidden":598},[622],[573,1467,1469,1472,1475,1478,1481,1484,1487],{"className":1468},[626],[573,1470],{"className":1471,"style":880},[630],[573,1473,848],{"className":1474},[635,670],[573,1476,852],{"className":1477},[887],[573,1479,1385],{"className":1480},[635,670],[573,1482],{"className":1483,"style":919},[643],[573,1485,1435],{"className":1486},[923],[573,1488],{"className":1489,"style":919},[643],[573,1491,1493,1496,1499,1502,1505,1508],{"className":1492},[626],[573,1494],{"className":1495,"style":880},[630],[573,1497,613],{"className":1498},[635,670],[573,1500,857],{"className":1501},[894],[573,1503],{"className":1504,"style":898},[643],[573,1506,860],{"className":1507},[902],[573,1509],{"className":1510,"style":898},[643],[573,1512,1514,1517,1520,1523,1526,1529,1532,1535],{"className":1513},[626],[573,1515],{"className":1516,"style":880},[630],[573,1518,848],{"className":1519},[635,670],[573,1521,852],{"className":1522},[887],[573,1524,1385],{"className":1525},[635,670],[573,1527,857],{"className":1528},[894],[573,1530],{"className":1531,"style":919},[643],[573,1533,1435],{"className":1534},[923],[573,1536],{"className":1537,"style":919},[643],[573,1539,1541,1544,1547,1550,1553],{"className":1540},[626],[573,1542],{"className":1543,"style":880},[630],[573,1545,848],{"className":1546},[635,670],[573,1548,852],{"className":1549},[887],[573,1551,613],{"className":1552},[635,670],[573,1554,857],{"className":1555},[894],[565,1557,1558,1559,1587,1588,1616,1617,1668,1669,1046],{},"Aus den beiden letzten Regeln folgt die Produktformel. Jeder Primteiler ",[573,1560,1562,1575],{"className":1561},[576],[573,1563,1565],{"className":1564},[580],[582,1566,1567],{"xmlns":584},[586,1568,1569,1573],{},[589,1570,1571],{},[611,1572,565],{},[615,1574,565],{"encoding":617},[573,1576,1578],{"className":1577,"ariaHidden":598},[622],[573,1579,1581,1584],{"className":1580},[626],[573,1582],{"className":1583,"style":829},[630],[573,1585,565],{"className":1586},[635,670]," kommt darin genau einmal vor, egal mit welcher Vielfachheit er in ",[573,1589,1591,1604],{"className":1590},[576],[573,1592,1594],{"className":1593},[580],[582,1595,1596],{"xmlns":584},[586,1597,1598,1602],{},[589,1599,1600],{},[611,1601,613],{},[615,1603,613],{"encoding":617},[573,1605,1607],{"className":1606,"ariaHidden":598},[622],[573,1608,1610,1613],{"className":1609},[626],[573,1611],{"className":1612,"style":697},[630],[573,1614,613],{"className":1615},[635,670]," steckt. Für ",[573,1618,1620,1638],{"className":1619},[576],[573,1621,1623],{"className":1622},[580],[582,1624,1625],{"xmlns":584},[586,1626,1627,1635],{},[589,1628,1629,1631,1633],{},[611,1630,613],{},[596,1632,860],{},[592,1634,594],{},[615,1636,1637],{"encoding":617},"n = 1",[573,1639,1641,1659],{"className":1640,"ariaHidden":598},[622],[573,1642,1644,1647,1650,1653,1656],{"className":1643},[626],[573,1645],{"className":1646,"style":697},[630],[573,1648,613],{"className":1649},[635,670],[573,1651],{"className":1652,"style":898},[643],[573,1654,860],{"className":1655},[902],[573,1657],{"className":1658,"style":898},[643],[573,1660,1662,1665],{"className":1661},[626],[573,1663],{"className":1664,"style":760},[630],[573,1666,594],{"className":1667},[635]," ist das Produkt leer, also ",[573,1670,1672,1696],{"className":1671},[576],[573,1673,1675],{"className":1674},[580],[582,1676,1677],{"xmlns":584},[586,1678,1679,1693],{},[589,1680,1681,1683,1685,1687,1689,1691],{},[611,1682,848],{},[596,1684,852],{"stretchy":851},[592,1686,594],{},[596,1688,857],{"stretchy":851},[596,1690,860],{},[592,1692,594],{},[615,1694,1695],{"encoding":617},"\\varphi(1) = 1",[573,1697,1699,1726],{"className":1698,"ariaHidden":598},[622],[573,1700,1702,1705,1708,1711,1714,1717,1720,1723],{"className":1701},[626],[573,1703],{"className":1704,"style":880},[630],[573,1706,848],{"className":1707},[635,670],[573,1709,852],{"className":1710},[887],[573,1712,594],{"className":1713},[635],[573,1715,857],{"className":1716},[894],[573,1718],{"className":1719,"style":898},[643],[573,1721,860],{"className":1722},[902],[573,1724],{"className":1725,"style":898},[643],[573,1727,1729,1732],{"className":1728},[626],[573,1730],{"className":1731,"style":760},[630],[573,1733,594],{"className":1734},[635],[560,1736,1738],{"id":1737},"beispiele","Beispiele",[565,1740,1741,1869],{},[573,1742,1744,1776],{"className":1743},[576],[573,1745,1747],{"className":1746},[580],[582,1748,1749],{"xmlns":584},[586,1750,1751,1773],{},[589,1752,1753,1755,1757,1760,1762,1768,1770],{},[611,1754,613],{},[596,1756,860],{},[592,1758,1759],{},"12",[596,1761,860],{},[1067,1763,1764,1766],{},[592,1765,602],{},[592,1767,602],{},[596,1769,1435],{},[592,1771,1772],{},"3",[615,1774,1775],{"encoding":617},"n = 12 = 2^2 \\cdot 3",[573,1777,1779,1797,1815,1860],{"className":1778,"ariaHidden":598},[622],[573,1780,1782,1785,1788,1791,1794],{"className":1781},[626],[573,1783],{"className":1784,"style":697},[630],[573,1786,613],{"className":1787},[635,670],[573,1789],{"className":1790,"style":898},[643],[573,1792,860],{"className":1793},[902],[573,1795],{"className":1796,"style":898},[643],[573,1798,1800,1803,1806,1809,1812],{"className":1799},[626],[573,1801],{"className":1802,"style":760},[630],[573,1804,1759],{"className":1805},[635],[573,1807],{"className":1808,"style":898},[643],[573,1810,860],{"className":1811},[902],[573,1813],{"className":1814,"style":898},[643],[573,1816,1818,1822,1851,1854,1857],{"className":1817},[626],[573,1819],{"className":1820,"style":1821},[630],"height:0.8141em;",[573,1823,1825,1828],{"className":1824},[635],[573,1826,602],{"className":1827},[635],[573,1829,1831],{"className":1830},[1126],[573,1832,1834],{"className":1833},[1130],[573,1835,1837],{"className":1836},[1134],[573,1838,1840],{"className":1839,"style":1821},[1138],[573,1841,1842,1845],{"style":1142},[573,1843],{"className":1844,"style":1147},[1146],[573,1846,1848],{"className":1847},[1151,1152,1153,1154],[573,1849,602],{"className":1850},[635,1154],[573,1852],{"className":1853,"style":919},[643],[573,1855,1435],{"className":1856},[923],[573,1858],{"className":1859,"style":919},[643],[573,1861,1863,1866],{"className":1862},[626],[573,1864],{"className":1865,"style":760},[630],[573,1867,1772],{"className":1868},[635],":",[573,1871,1874],{"className":1872},[1873],"katex-display",[573,1875,1877,1950],{"className":1876},[576],[573,1878,1880],{"className":1879},[580],[582,1881,1883],{"xmlns":584,"display":1882},"block",[586,1884,1885,1947],{},[589,1886,1887,1889,1891,1893,1895,1897,1899,1901,1922,1924,1942,1944],{},[611,1888,848],{},[596,1890,852],{"stretchy":851},[592,1892,1759],{},[596,1894,857],{"stretchy":851},[596,1896,860],{},[592,1898,1759],{},[596,1900,1435],{},[589,1902,1903,1905,1907,1909,1920],{},[596,1904,852],{"fence":598},[592,1906,594],{},[596,1908,865],{},[1910,1911,1913],"mstyle",{"scriptlevel":1912,"displaystyle":851},"0",[1914,1915,1916,1918],"mfrac",{},[592,1917,594],{},[592,1919,602],{},[596,1921,857],{"fence":598},[596,1923,1435],{},[589,1925,1926,1928,1930,1932,1940],{},[596,1927,852],{"fence":598},[592,1929,594],{},[596,1931,865],{},[1910,1933,1934],{"scriptlevel":1912,"displaystyle":851},[1914,1935,1936,1938],{},[592,1937,594],{},[592,1939,1772],{},[596,1941,857],{"fence":598},[596,1943,860],{},[592,1945,1946],{},"4",[615,1948,1949],{"encoding":617},"\\varphi(12) = 12 \\cdot \\left(1 - \\tfrac{1}{2}\\right) \\cdot \\left(1 - \\tfrac{1}{3}\\right) = 4",[573,1951,1953,1980,1998,2125,2235],{"className":1952,"ariaHidden":598},[622],[573,1954,1956,1959,1962,1965,1968,1971,1974,1977],{"className":1955},[626],[573,1957],{"className":1958,"style":880},[630],[573,1960,848],{"className":1961},[635,670],[573,1963,852],{"className":1964},[887],[573,1966,1759],{"className":1967},[635],[573,1969,857],{"className":1970},[894],[573,1972],{"className":1973,"style":898},[643],[573,1975,860],{"className":1976},[902],[573,1978],{"className":1979,"style":898},[643],[573,1981,1983,1986,1989,1992,1995],{"className":1982},[626],[573,1984],{"className":1985,"style":760},[630],[573,1987,1759],{"className":1988},[635],[573,1990],{"className":1991,"style":919},[643],[573,1993,1435],{"className":1994},[923],[573,1996],{"className":1997,"style":919},[643],[573,1999,2001,2005,2116,2119,2122],{"className":2000},[626],[573,2002],{"className":2003,"style":2004},[630],"height:1.2em;vertical-align:-0.35em;",[573,2006,2008,2018,2021,2024,2027,2030,2110],{"className":2007},[657],[573,2009,2013],{"className":2010,"style":2012},[887,2011],"delimcenter","top:0em;",[573,2014,852],{"className":2015},[2016,2017],"delimsizing","size1",[573,2019,594],{"className":2020},[635],[573,2022],{"className":2023,"style":919},[643],[573,2025,865],{"className":2026},[923],[573,2028],{"className":2029,"style":919},[643],[573,2031,2033,2037,2107],{"className":2032},[635],[573,2034],{"className":2035},[887,2036],"nulldelimiter",[573,2038,2040],{"className":2039},[1914],[573,2041,2044,2098],{"className":2042},[1130,2043],"vlist-t2",[573,2045,2047,2093],{"className":2046},[1134],[573,2048,2051,2067,2078],{"className":2049,"style":2050},[1138],"height:0.8451em;",[573,2052,2054,2058],{"style":2053},"top:-2.655em;",[573,2055],{"className":2056,"style":2057},[1146],"height:3em;",[573,2059,2061],{"className":2060},[1151,1152,1153,1154],[573,2062,2064],{"className":2063},[635,1154],[573,2065,602],{"className":2066},[635,1154],[573,2068,2070,2073],{"style":2069},"top:-3.23em;",[573,2071],{"className":2072,"style":2057},[1146],[573,2074],{"className":2075,"style":2077},[2076],"frac-line","border-bottom-width:0.04em;",[573,2079,2081,2084],{"style":2080},"top:-3.394em;",[573,2082],{"className":2083,"style":2057},[1146],[573,2085,2087],{"className":2086},[1151,1152,1153,1154],[573,2088,2090],{"className":2089},[635,1154],[573,2091,594],{"className":2092},[635,1154],[573,2094,2097],{"className":2095},[2096],"vlist-s","​",[573,2099,2101],{"className":2100},[1134],[573,2102,2105],{"className":2103,"style":2104},[1138],"height:0.345em;",[573,2106],{},[573,2108],{"className":2109},[894,2036],[573,2111,2113],{"className":2112,"style":2012},[894,2011],[573,2114,857],{"className":2115},[2016,2017],[573,2117],{"className":2118,"style":919},[643],[573,2120,1435],{"className":2121},[923],[573,2123],{"className":2124,"style":919},[643],[573,2126,2128,2131,2226,2229,2232],{"className":2127},[626],[573,2129],{"className":2130,"style":2004},[630],[573,2132,2134,2140,2143,2146,2149,2152,2220],{"className":2133},[657],[573,2135,2137],{"className":2136,"style":2012},[887,2011],[573,2138,852],{"className":2139},[2016,2017],[573,2141,594],{"className":2142},[635],[573,2144],{"className":2145,"style":919},[643],[573,2147,865],{"className":2148},[923],[573,2150],{"className":2151,"style":919},[643],[573,2153,2155,2158,2217],{"className":2154},[635],[573,2156],{"className":2157},[887,2036],[573,2159,2161],{"className":2160},[1914],[573,2162,2164,2209],{"className":2163},[1130,2043],[573,2165,2167,2206],{"className":2166},[1134],[573,2168,2170,2184,2192],{"className":2169,"style":2050},[1138],[573,2171,2172,2175],{"style":2053},[573,2173],{"className":2174,"style":2057},[1146],[573,2176,2178],{"className":2177},[1151,1152,1153,1154],[573,2179,2181],{"className":2180},[635,1154],[573,2182,1772],{"className":2183},[635,1154],[573,2185,2186,2189],{"style":2069},[573,2187],{"className":2188,"style":2057},[1146],[573,2190],{"className":2191,"style":2077},[2076],[573,2193,2194,2197],{"style":2080},[573,2195],{"className":2196,"style":2057},[1146],[573,2198,2200],{"className":2199},[1151,1152,1153,1154],[573,2201,2203],{"className":2202},[635,1154],[573,2204,594],{"className":2205},[635,1154],[573,2207,2097],{"className":2208},[2096],[573,2210,2212],{"className":2211},[1134],[573,2213,2215],{"className":2214,"style":2104},[1138],[573,2216],{},[573,2218],{"className":2219},[894,2036],[573,2221,2223],{"className":2222,"style":2012},[894,2011],[573,2224,857],{"className":2225},[2016,2017],[573,2227],{"className":2228,"style":898},[643],[573,2230,860],{"className":2231},[902],[573,2233],{"className":2234,"style":898},[643],[573,2236,2238,2241],{"className":2237},[626],[573,2239],{"className":2240,"style":760},[630],[573,2242,1946],{"className":2243},[635],[565,2245,2246,2247,1046],{},"Die teilerfremden Zahlen sind ",[573,2248,2250,2279],{"className":2249},[576],[573,2251,2253],{"className":2252},[580],[582,2254,2255],{"xmlns":584},[586,2256,2257,2276],{},[589,2258,2259,2261,2263,2266,2268,2271,2273],{},[592,2260,594],{},[596,2262,599],{"separator":598},[592,2264,2265],{},"5",[596,2267,599],{"separator":598},[592,2269,2270],{},"7",[596,2272,599],{"separator":598},[592,2274,2275],{},"11",[615,2277,2278],{"encoding":617},"1, 5, 7, 11",[573,2280,2282],{"className":2281,"ariaHidden":598},[622],[573,2283,2285,2288,2291,2294,2297,2300,2303,2306,2309,2312,2315],{"className":2284},[626],[573,2286],{"className":2287,"style":631},[630],[573,2289,594],{"className":2290},[635],[573,2292,599],{"className":2293},[639],[573,2295],{"className":2296,"style":644},[643],[573,2298,2265],{"className":2299},[635],[573,2301,599],{"className":2302},[639],[573,2304],{"className":2305,"style":644},[643],[573,2307,2270],{"className":2308},[635],[573,2310,599],{"className":2311},[639],[573,2313],{"className":2314,"style":644},[643],[573,2316,2275],{"className":2317},[635],[565,2319,2320,833,2476,1046],{},[573,2321,2323,2358],{"className":2322},[576],[573,2324,2326],{"className":2325},[580],[582,2327,2328],{"xmlns":584},[586,2329,2330,2355],{},[589,2331,2332,2334,2336,2339,2341,2347,2349],{},[611,2333,613],{},[596,2335,860],{},[592,2337,2338],{},"100",[596,2340,860],{},[1067,2342,2343,2345],{},[592,2344,602],{},[592,2346,602],{},[596,2348,1435],{},[1067,2350,2351,2353],{},[592,2352,2265],{},[592,2354,602],{},[615,2356,2357],{"encoding":617},"n = 100 = 2^2 \\cdot 5^2",[573,2359,2361,2379,2397,2441],{"className":2360,"ariaHidden":598},[622],[573,2362,2364,2367,2370,2373,2376],{"className":2363},[626],[573,2365],{"className":2366,"style":697},[630],[573,2368,613],{"className":2369},[635,670],[573,2371],{"className":2372,"style":898},[643],[573,2374,860],{"className":2375},[902],[573,2377],{"className":2378,"style":898},[643],[573,2380,2382,2385,2388,2391,2394],{"className":2381},[626],[573,2383],{"className":2384,"style":760},[630],[573,2386,2338],{"className":2387},[635],[573,2389],{"className":2390,"style":898},[643],[573,2392,860],{"className":2393},[902],[573,2395],{"className":2396,"style":898},[643],[573,2398,2400,2403,2432,2435,2438],{"className":2399},[626],[573,2401],{"className":2402,"style":1821},[630],[573,2404,2406,2409],{"className":2405},[635],[573,2407,602],{"className":2408},[635],[573,2410,2412],{"className":2411},[1126],[573,2413,2415],{"className":2414},[1130],[573,2416,2418],{"className":2417},[1134],[573,2419,2421],{"className":2420,"style":1821},[1138],[573,2422,2423,2426],{"style":1142},[573,2424],{"className":2425,"style":1147},[1146],[573,2427,2429],{"className":2428},[1151,1152,1153,1154],[573,2430,602],{"className":2431},[635,1154],[573,2433],{"className":2434,"style":919},[643],[573,2436,1435],{"className":2437},[923],[573,2439],{"className":2440,"style":919},[643],[573,2442,2444,2447],{"className":2443},[626],[573,2445],{"className":2446,"style":1821},[630],[573,2448,2450,2453],{"className":2449},[635],[573,2451,2265],{"className":2452},[635],[573,2454,2456],{"className":2455},[1126],[573,2457,2459],{"className":2458},[1130],[573,2460,2462],{"className":2461},[1134],[573,2463,2465],{"className":2464,"style":1821},[1138],[573,2466,2467,2470],{"style":1142},[573,2468],{"className":2469,"style":1147},[1146],[573,2471,2473],{"className":2472},[1151,1152,1153,1154],[573,2474,602],{"className":2475},[635,1154],[573,2477,2479,2524],{"className":2478},[576],[573,2480,2482],{"className":2481},[580],[582,2483,2484],{"xmlns":584},[586,2485,2486,2521],{},[589,2487,2488,2490,2492,2494,2496,2498,2500,2502,2508,2510,2516,2518],{},[611,2489,848],{},[596,2491,852],{"stretchy":851},[592,2493,2338],{},[596,2495,857],{"stretchy":851},[596,2497,860],{},[592,2499,2338],{},[596,2501,1435],{},[1914,2503,2504,2506],{},[592,2505,594],{},[592,2507,602],{},[596,2509,1435],{},[1914,2511,2512,2514],{},[592,2513,1946],{},[592,2515,2265],{},[596,2517,860],{},[592,2519,2520],{},"40",[615,2522,2523],{"encoding":617},"\\varphi(100) = 100 \\cdot \\frac{1}{2} \\cdot \\frac{4}{5} = 40",[573,2525,2527,2554,2572,2656,2739],{"className":2526,"ariaHidden":598},[622],[573,2528,2530,2533,2536,2539,2542,2545,2548,2551],{"className":2529},[626],[573,2531],{"className":2532,"style":880},[630],[573,2534,848],{"className":2535},[635,670],[573,2537,852],{"className":2538},[887],[573,2540,2338],{"className":2541},[635],[573,2543,857],{"className":2544},[894],[573,2546],{"className":2547,"style":898},[643],[573,2549,860],{"className":2550},[902],[573,2552],{"className":2553,"style":898},[643],[573,2555,2557,2560,2563,2566,2569],{"className":2556},[626],[573,2558],{"className":2559,"style":760},[630],[573,2561,2338],{"className":2562},[635],[573,2564],{"className":2565,"style":919},[643],[573,2567,1435],{"className":2568},[923],[573,2570],{"className":2571,"style":919},[643],[573,2573,2575,2579,2647,2650,2653],{"className":2574},[626],[573,2576],{"className":2577,"style":2578},[630],"height:1.1901em;vertical-align:-0.345em;",[573,2580,2582,2585,2644],{"className":2581},[635],[573,2583],{"className":2584},[887,2036],[573,2586,2588],{"className":2587},[1914],[573,2589,2591,2636],{"className":2590},[1130,2043],[573,2592,2594,2633],{"className":2593},[1134],[573,2595,2597,2611,2619],{"className":2596,"style":2050},[1138],[573,2598,2599,2602],{"style":2053},[573,2600],{"className":2601,"style":2057},[1146],[573,2603,2605],{"className":2604},[1151,1152,1153,1154],[573,2606,2608],{"className":2607},[635,1154],[573,2609,602],{"className":2610},[635,1154],[573,2612,2613,2616],{"style":2069},[573,2614],{"className":2615,"style":2057},[1146],[573,2617],{"className":2618,"style":2077},[2076],[573,2620,2621,2624],{"style":2080},[573,2622],{"className":2623,"style":2057},[1146],[573,2625,2627],{"className":2626},[1151,1152,1153,1154],[573,2628,2630],{"className":2629},[635,1154],[573,2631,594],{"className":2632},[635,1154],[573,2634,2097],{"className":2635},[2096],[573,2637,2639],{"className":2638},[1134],[573,2640,2642],{"className":2641,"style":2104},[1138],[573,2643],{},[573,2645],{"className":2646},[894,2036],[573,2648],{"className":2649,"style":919},[643],[573,2651,1435],{"className":2652},[923],[573,2654],{"className":2655,"style":919},[643],[573,2657,2659,2662,2730,2733,2736],{"className":2658},[626],[573,2660],{"className":2661,"style":2578},[630],[573,2663,2665,2668,2727],{"className":2664},[635],[573,2666],{"className":2667},[887,2036],[573,2669,2671],{"className":2670},[1914],[573,2672,2674,2719],{"className":2673},[1130,2043],[573,2675,2677,2716],{"className":2676},[1134],[573,2678,2680,2694,2702],{"className":2679,"style":2050},[1138],[573,2681,2682,2685],{"style":2053},[573,2683],{"className":2684,"style":2057},[1146],[573,2686,2688],{"className":2687},[1151,1152,1153,1154],[573,2689,2691],{"className":2690},[635,1154],[573,2692,2265],{"className":2693},[635,1154],[573,2695,2696,2699],{"style":2069},[573,2697],{"className":2698,"style":2057},[1146],[573,2700],{"className":2701,"style":2077},[2076],[573,2703,2704,2707],{"style":2080},[573,2705],{"className":2706,"style":2057},[1146],[573,2708,2710],{"className":2709},[1151,1152,1153,1154],[573,2711,2713],{"className":2712},[635,1154],[573,2714,1946],{"className":2715},[635,1154],[573,2717,2097],{"className":2718},[2096],[573,2720,2722],{"className":2721},[1134],[573,2723,2725],{"className":2724,"style":2104},[1138],[573,2726],{},[573,2728],{"className":2729},[894,2036],[573,2731],{"className":2732,"style":898},[643],[573,2734,860],{"className":2735},[902],[573,2737],{"className":2738,"style":898},[643],[573,2740,2742,2745],{"className":2741},[626],[573,2743],{"className":2744,"style":760},[630],[573,2746,2520],{"className":2747},[635],[565,2749,2750,2751,833,2833,3005,3006,833,3035,1046],{},"Primzahlpotenz ",[573,2752,2754,2777],{"className":2753},[576],[573,2755,2757],{"className":2756},[580],[582,2758,2759],{"xmlns":584},[586,2760,2761,2774],{},[589,2762,2763,2766,2768],{},[592,2764,2765],{},"81",[596,2767,860],{},[1067,2769,2770,2772],{},[592,2771,1772],{},[592,2773,1946],{},[615,2775,2776],{"encoding":617},"81 = 3^4",[573,2778,2780,2798],{"className":2779,"ariaHidden":598},[622],[573,2781,2783,2786,2789,2792,2795],{"className":2782},[626],[573,2784],{"className":2785,"style":760},[630],[573,2787,2765],{"className":2788},[635],[573,2790],{"className":2791,"style":898},[643],[573,2793,860],{"className":2794},[902],[573,2796],{"className":2797,"style":898},[643],[573,2799,2801,2804],{"className":2800},[626],[573,2802],{"className":2803,"style":1821},[630],[573,2805,2807,2810],{"className":2806},[635],[573,2808,1772],{"className":2809},[635],[573,2811,2813],{"className":2812},[1126],[573,2814,2816],{"className":2815},[1130],[573,2817,2819],{"className":2818},[1134],[573,2820,2822],{"className":2821,"style":1821},[1138],[573,2823,2824,2827],{"style":1142},[573,2825],{"className":2826,"style":1147},[1146],[573,2828,2830],{"className":2829},[1151,1152,1153,1154],[573,2831,1946],{"className":2832},[635,1154],[573,2834,2836,2877],{"className":2835},[576],[573,2837,2839],{"className":2838},[580],[582,2840,2841],{"xmlns":584},[586,2842,2843,2874],{},[589,2844,2845,2847,2849,2851,2853,2855,2861,2863,2869,2871],{},[611,2846,848],{},[596,2848,852],{"stretchy":851},[592,2850,2765],{},[596,2852,857],{"stretchy":851},[596,2854,860],{},[1067,2856,2857,2859],{},[592,2858,1772],{},[592,2860,1946],{},[596,2862,865],{},[1067,2864,2865,2867],{},[592,2866,1772],{},[592,2868,1772],{},[596,2870,860],{},[592,2872,2873],{},"54",[615,2875,2876],{"encoding":617},"\\varphi(81) = 3^4 - 3^3 = 54",[573,2878,2880,2907,2952,2996],{"className":2879,"ariaHidden":598},[622],[573,2881,2883,2886,2889,2892,2895,2898,2901,2904],{"className":2882},[626],[573,2884],{"className":2885,"style":880},[630],[573,2887,848],{"className":2888},[635,670],[573,2890,852],{"className":2891},[887],[573,2893,2765],{"className":2894},[635],[573,2896,857],{"className":2897},[894],[573,2899],{"className":2900,"style":898},[643],[573,2902,860],{"className":2903},[902],[573,2905],{"className":2906,"style":898},[643],[573,2908,2910,2914,2943,2946,2949],{"className":2909},[626],[573,2911],{"className":2912,"style":2913},[630],"height:0.8974em;vertical-align:-0.0833em;",[573,2915,2917,2920],{"className":2916},[635],[573,2918,1772],{"className":2919},[635],[573,2921,2923],{"className":2922},[1126],[573,2924,2926],{"className":2925},[1130],[573,2927,2929],{"className":2928},[1134],[573,2930,2932],{"className":2931,"style":1821},[1138],[573,2933,2934,2937],{"style":1142},[573,2935],{"className":2936,"style":1147},[1146],[573,2938,2940],{"className":2939},[1151,1152,1153,1154],[573,2941,1946],{"className":2942},[635,1154],[573,2944],{"className":2945,"style":919},[643],[573,2947,865],{"className":2948},[923],[573,2950],{"className":2951,"style":919},[643],[573,2953,2955,2958,2987,2990,2993],{"className":2954},[626],[573,2956],{"className":2957,"style":1821},[630],[573,2959,2961,2964],{"className":2960},[635],[573,2962,1772],{"className":2963},[635],[573,2965,2967],{"className":2966},[1126],[573,2968,2970],{"className":2969},[1130],[573,2971,2973],{"className":2972},[1134],[573,2974,2976],{"className":2975,"style":1821},[1138],[573,2977,2978,2981],{"style":1142},[573,2979],{"className":2980,"style":1147},[1146],[573,2982,2984],{"className":2983},[1151,1152,1153,1154],[573,2985,1772],{"className":2986},[635,1154],[573,2988],{"className":2989,"style":898},[643],[573,2991,860],{"className":2992},[902],[573,2994],{"className":2995,"style":898},[643],[573,2997,2999,3002],{"className":2998},[626],[573,3000],{"className":3001,"style":760},[630],[573,3003,2873],{"className":3004},[635],". Primzahl ",[573,3007,3009,3023],{"className":3008},[576],[573,3010,3012],{"className":3011},[580],[582,3013,3014],{"xmlns":584},[586,3015,3016,3021],{},[589,3017,3018],{},[592,3019,3020],{},"13",[615,3022,3020],{"encoding":617},[573,3024,3026],{"className":3025,"ariaHidden":598},[622],[573,3027,3029,3032],{"className":3028},[626],[573,3030],{"className":3031,"style":760},[630],[573,3033,3020],{"className":3034},[635],[573,3036,3038,3062],{"className":3037},[576],[573,3039,3041],{"className":3040},[580],[582,3042,3043],{"xmlns":584},[586,3044,3045,3059],{},[589,3046,3047,3049,3051,3053,3055,3057],{},[611,3048,848],{},[596,3050,852],{"stretchy":851},[592,3052,3020],{},[596,3054,857],{"stretchy":851},[596,3056,860],{},[592,3058,1759],{},[615,3060,3061],{"encoding":617},"\\varphi(13) = 12",[573,3063,3065,3092],{"className":3064,"ariaHidden":598},[622],[573,3066,3068,3071,3074,3077,3080,3083,3086,3089],{"className":3067},[626],[573,3069],{"className":3070,"style":880},[630],[573,3072,848],{"className":3073},[635,670],[573,3075,852],{"className":3076},[887],[573,3078,3020],{"className":3079},[635],[573,3081,857],{"className":3082},[894],[573,3084],{"className":3085,"style":898},[643],[573,3087,860],{"className":3088},[902],[573,3090],{"className":3091,"style":898},[643],[573,3093,3095,3098],{"className":3094},[626],[573,3096],{"className":3097,"style":760},[630],[573,3099,1759],{"className":3100},[635],[565,3102,3103,3104,1046],{},"Multiplikativität: ",[573,3105,3107,3162],{"className":3106},[576],[573,3108,3110],{"className":3109},[580],[582,3111,3112],{"xmlns":584},[586,3113,3114,3159],{},[589,3115,3116,3118,3120,3123,3125,3127,3129,3131,3133,3135,3137,3139,3141,3144,3146,3148,3150,3152,3155,3157],{},[611,3117,848],{},[596,3119,852],{"stretchy":851},[592,3121,3122],{},"36",[596,3124,857],{"stretchy":851},[596,3126,860],{},[611,3128,848],{},[596,3130,852],{"stretchy":851},[592,3132,1946],{},[596,3134,857],{"stretchy":851},[596,3136,1435],{},[611,3138,848],{},[596,3140,852],{"stretchy":851},[592,3142,3143],{},"9",[596,3145,857],{"stretchy":851},[596,3147,860],{},[592,3149,602],{},[596,3151,1435],{},[592,3153,3154],{},"6",[596,3156,860],{},[592,3158,1759],{},[615,3160,3161],{"encoding":617},"\\varphi(36) = \\varphi(4) \\cdot \\varphi(9) = 2 \\cdot 6 = 12",[573,3163,3165,3192,3219,3246,3264,3282],{"className":3164,"ariaHidden":598},[622],[573,3166,3168,3171,3174,3177,3180,3183,3186,3189],{"className":3167},[626],[573,3169],{"className":3170,"style":880},[630],[573,3172,848],{"className":3173},[635,670],[573,3175,852],{"className":3176},[887],[573,3178,3122],{"className":3179},[635],[573,3181,857],{"className":3182},[894],[573,3184],{"className":3185,"style":898},[643],[573,3187,860],{"className":3188},[902],[573,3190],{"className":3191,"style":898},[643],[573,3193,3195,3198,3201,3204,3207,3210,3213,3216],{"className":3194},[626],[573,3196],{"className":3197,"style":880},[630],[573,3199,848],{"className":3200},[635,670],[573,3202,852],{"className":3203},[887],[573,3205,1946],{"className":3206},[635],[573,3208,857],{"className":3209},[894],[573,3211],{"className":3212,"style":919},[643],[573,3214,1435],{"className":3215},[923],[573,3217],{"className":3218,"style":919},[643],[573,3220,3222,3225,3228,3231,3234,3237,3240,3243],{"className":3221},[626],[573,3223],{"className":3224,"style":880},[630],[573,3226,848],{"className":3227},[635,670],[573,3229,852],{"className":3230},[887],[573,3232,3143],{"className":3233},[635],[573,3235,857],{"className":3236},[894],[573,3238],{"className":3239,"style":898},[643],[573,3241,860],{"className":3242},[902],[573,3244],{"className":3245,"style":898},[643],[573,3247,3249,3252,3255,3258,3261],{"className":3248},[626],[573,3250],{"className":3251,"style":760},[630],[573,3253,602],{"className":3254},[635],[573,3256],{"className":3257,"style":919},[643],[573,3259,1435],{"className":3260},[923],[573,3262],{"className":3263,"style":919},[643],[573,3265,3267,3270,3273,3276,3279],{"className":3266},[626],[573,3268],{"className":3269,"style":760},[630],[573,3271,3154],{"className":3272},[635],[573,3274],{"className":3275,"style":898},[643],[573,3277,860],{"className":3278},[902],[573,3280],{"className":3281,"style":898},[643],[573,3283,3285,3288],{"className":3284},[626],[573,3286],{"className":3287,"style":760},[630],[573,3289,1759],{"className":3290},[635],[560,3292,3294],{"id":3293},"typische-fehler","Typische Fehler",[798,3296,3297,3531,3767],{},[801,3298,3299,701,3302,3368,3369,3397,3398,3426,3427,3530],{},[569,3300,3301],{},"Multiplikativität ohne Teilerfremdheit anwenden.",[573,3303,3305,3329],{"className":3304},[576],[573,3306,3308],{"className":3307},[580],[582,3309,3310],{"xmlns":584},[586,3311,3312,3326],{},[589,3313,3314,3316,3318,3320,3322,3324],{},[611,3315,848],{},[596,3317,852],{"stretchy":851},[592,3319,1946],{},[596,3321,857],{"stretchy":851},[596,3323,860],{},[592,3325,602],{},[615,3327,3328],{"encoding":617},"\\varphi(4) = 2",[573,3330,3332,3359],{"className":3331,"ariaHidden":598},[622],[573,3333,3335,3338,3341,3344,3347,3350,3353,3356],{"className":3334},[626],[573,3336],{"className":3337,"style":880},[630],[573,3339,848],{"className":3340},[635,670],[573,3342,852],{"className":3343},[887],[573,3345,1946],{"className":3346},[635],[573,3348,857],{"className":3349},[894],[573,3351],{"className":3352,"style":898},[643],[573,3354,860],{"className":3355},[902],[573,3357],{"className":3358,"style":898},[643],[573,3360,3362,3365],{"className":3361},[626],[573,3363],{"className":3364,"style":760},[630],[573,3366,602],{"className":3367},[635]," (nämlich ",[573,3370,3372,3385],{"className":3371},[576],[573,3373,3375],{"className":3374},[580],[582,3376,3377],{"xmlns":584},[586,3378,3379,3383],{},[589,3380,3381],{},[592,3382,594],{},[615,3384,594],{"encoding":617},[573,3386,3388],{"className":3387,"ariaHidden":598},[622],[573,3389,3391,3394],{"className":3390},[626],[573,3392],{"className":3393,"style":760},[630],[573,3395,594],{"className":3396},[635]," und ",[573,3399,3401,3414],{"className":3400},[576],[573,3402,3404],{"className":3403},[580],[582,3405,3406],{"xmlns":584},[586,3407,3408,3412],{},[589,3409,3410],{},[592,3411,1772],{},[615,3413,1772],{"encoding":617},[573,3415,3417],{"className":3416,"ariaHidden":598},[622],[573,3418,3420,3423],{"className":3419},[626],[573,3421],{"className":3422,"style":760},[630],[573,3424,1772],{"className":3425},[635],"), aber ",[573,3428,3430,3464],{"className":3429},[576],[573,3431,3433],{"className":3432},[580],[582,3434,3435],{"xmlns":584},[586,3436,3437,3461],{},[589,3438,3439,3441,3443,3445,3447,3449,3451,3453,3455,3457,3459],{},[611,3440,848],{},[596,3442,852],{"stretchy":851},[592,3444,602],{},[596,3446,857],{"stretchy":851},[596,3448,1435],{},[611,3450,848],{},[596,3452,852],{"stretchy":851},[592,3454,602],{},[596,3456,857],{"stretchy":851},[596,3458,860],{},[592,3460,594],{},[615,3462,3463],{"encoding":617},"\\varphi(2) \\cdot \\varphi(2) = 1",[573,3465,3467,3494,3521],{"className":3466,"ariaHidden":598},[622],[573,3468,3470,3473,3476,3479,3482,3485,3488,3491],{"className":3469},[626],[573,3471],{"className":3472,"style":880},[630],[573,3474,848],{"className":3475},[635,670],[573,3477,852],{"className":3478},[887],[573,3480,602],{"className":3481},[635],[573,3483,857],{"className":3484},[894],[573,3486],{"className":3487,"style":919},[643],[573,3489,1435],{"className":3490},[923],[573,3492],{"className":3493,"style":919},[643],[573,3495,3497,3500,3503,3506,3509,3512,3515,3518],{"className":3496},[626],[573,3498],{"className":3499,"style":880},[630],[573,3501,848],{"className":3502},[635,670],[573,3504,852],{"className":3505},[887],[573,3507,602],{"className":3508},[635],[573,3510,857],{"className":3511},[894],[573,3513],{"className":3514,"style":898},[643],[573,3516,860],{"className":3517},[902],[573,3519],{"className":3520,"style":898},[643],[573,3522,3524,3527],{"className":3523},[626],[573,3525],{"className":3526,"style":760},[630],[573,3528,594],{"className":3529},[635],". Die Regel gilt nur, wenn die Faktoren teilerfremd sind.",[801,3532,3533,3536,3537,3632,3633,3766],{},[569,3534,3535],{},"Primfaktoren mehrfach einsetzen."," Bei ",[573,3538,3540,3566],{"className":3539},[576],[573,3541,3543],{"className":3542},[580],[582,3544,3545],{"xmlns":584},[586,3546,3547,3563],{},[589,3548,3549,3551,3553,3555,3557,3559,3561],{},[592,3550,1759],{},[596,3552,860],{},[592,3554,602],{},[596,3556,1435],{},[592,3558,602],{},[596,3560,1435],{},[592,3562,1772],{},[615,3564,3565],{"encoding":617},"12 = 2 \\cdot 2 \\cdot 3",[573,3567,3569,3587,3605,3623],{"className":3568,"ariaHidden":598},[622],[573,3570,3572,3575,3578,3581,3584],{"className":3571},[626],[573,3573],{"className":3574,"style":760},[630],[573,3576,1759],{"className":3577},[635],[573,3579],{"className":3580,"style":898},[643],[573,3582,860],{"className":3583},[902],[573,3585],{"className":3586,"style":898},[643],[573,3588,3590,3593,3596,3599,3602],{"className":3589},[626],[573,3591],{"className":3592,"style":760},[630],[573,3594,602],{"className":3595},[635],[573,3597],{"className":3598,"style":919},[643],[573,3600,1435],{"className":3601},[923],[573,3603],{"className":3604,"style":919},[643],[573,3606,3608,3611,3614,3617,3620],{"className":3607},[626],[573,3609],{"className":3610,"style":760},[630],[573,3612,602],{"className":3613},[635],[573,3615],{"className":3616,"style":919},[643],[573,3618,1435],{"className":3619},[923],[573,3621],{"className":3622,"style":919},[643],[573,3624,3626,3629],{"className":3625},[626],[573,3627],{"className":3628,"style":760},[630],[573,3630,1772],{"className":3631},[635]," wird der Faktor ",[573,3634,3636,3662],{"className":3635},[576],[573,3637,3639],{"className":3638},[580],[582,3640,3641],{"xmlns":584},[586,3642,3643,3659],{},[589,3644,3645,3647,3649,3651,3657],{},[596,3646,852],{"fence":598},[592,3648,594],{},[596,3650,865],{},[1914,3652,3653,3655],{},[592,3654,594],{},[592,3656,602],{},[596,3658,857],{"fence":598},[615,3660,3661],{"encoding":617},"\\left(1 - \\frac{1}{2}\\right)",[573,3663,3665],{"className":3664,"ariaHidden":598},[622],[573,3666,3668,3671],{"className":3667},[626],[573,3669],{"className":3670,"style":2004},[630],[573,3672,3674,3680,3683,3686,3689,3692,3760],{"className":3673},[657],[573,3675,3677],{"className":3676,"style":2012},[887,2011],[573,3678,852],{"className":3679},[2016,2017],[573,3681,594],{"className":3682},[635],[573,3684],{"className":3685,"style":919},[643],[573,3687,865],{"className":3688},[923],[573,3690],{"className":3691,"style":919},[643],[573,3693,3695,3698,3757],{"className":3694},[635],[573,3696],{"className":3697},[887,2036],[573,3699,3701],{"className":3700},[1914],[573,3702,3704,3749],{"className":3703},[1130,2043],[573,3705,3707,3746],{"className":3706},[1134],[573,3708,3710,3724,3732],{"className":3709,"style":2050},[1138],[573,3711,3712,3715],{"style":2053},[573,3713],{"className":3714,"style":2057},[1146],[573,3716,3718],{"className":3717},[1151,1152,1153,1154],[573,3719,3721],{"className":3720},[635,1154],[573,3722,602],{"className":3723},[635,1154],[573,3725,3726,3729],{"style":2069},[573,3727],{"className":3728,"style":2057},[1146],[573,3730],{"className":3731,"style":2077},[2076],[573,3733,3734,3737],{"style":2080},[573,3735],{"className":3736,"style":2057},[1146],[573,3738,3740],{"className":3739},[1151,1152,1153,1154],[573,3741,3743],{"className":3742},[635,1154],[573,3744,594],{"className":3745},[635,1154],[573,3747,2097],{"className":3748},[2096],[573,3750,3752],{"className":3751},[1134],[573,3753,3755],{"className":3754,"style":2104},[1138],[573,3756],{},[573,3758],{"className":3759},[894,2036],[573,3761,3763],{"className":3762,"style":2012},[894,2011],[573,3764,857],{"className":3765},[2016,2017]," nur einmal genommen.",[801,3768,3769,3890,3891,3957,3958,1046],{},[569,3770,3771,3860,3861,3889],{},[573,3772,3774,3802],{"className":3773},[576],[573,3775,3777],{"className":3776},[580],[582,3778,3779],{"xmlns":584},[586,3780,3781,3799],{},[589,3782,3783,3785,3787,3789,3791,3793,3795,3797],{},[611,3784,848],{},[596,3786,852],{"stretchy":851},[611,3788,613],{},[596,3790,857],{"stretchy":851},[596,3792,860],{},[611,3794,613],{},[596,3796,865],{},[592,3798,594],{},[615,3800,3801],{"encoding":617},"\\varphi(n) = n - 1",[573,3803,3805,3832,3851],{"className":3804,"ariaHidden":598},[622],[573,3806,3808,3811,3814,3817,3820,3823,3826,3829],{"className":3807},[626],[573,3809],{"className":3810,"style":880},[630],[573,3812,848],{"className":3813},[635,670],[573,3815,852],{"className":3816},[887],[573,3818,613],{"className":3819},[635,670],[573,3821,857],{"className":3822},[894],[573,3824],{"className":3825,"style":898},[643],[573,3827,860],{"className":3828},[902],[573,3830],{"className":3831,"style":898},[643],[573,3833,3835,3839,3842,3845,3848],{"className":3834},[626],[573,3836],{"className":3837,"style":3838},[630],"height:0.6667em;vertical-align:-0.0833em;",[573,3840,613],{"className":3841},[635,670],[573,3843],{"className":3844,"style":919},[643],[573,3846,865],{"className":3847},[923],[573,3849],{"className":3850,"style":919},[643],[573,3852,3854,3857],{"className":3853},[626],[573,3855],{"className":3856,"style":760},[630],[573,3858,594],{"className":3859},[635]," für jedes ",[573,3862,3864,3877],{"className":3863},[576],[573,3865,3867],{"className":3866},[580],[582,3868,3869],{"xmlns":584},[586,3870,3871,3875],{},[589,3872,3873],{},[611,3874,613],{},[615,3876,613],{"encoding":617},[573,3878,3880],{"className":3879,"ariaHidden":598},[622],[573,3881,3883,3886],{"className":3882},[626],[573,3884],{"className":3885,"style":697},[630],[573,3887,613],{"className":3888},[635,670]," annehmen."," Das gilt nur für Primzahlen; ",[573,3892,3894,3918],{"className":3893},[576],[573,3895,3897],{"className":3896},[580],[582,3898,3899],{"xmlns":584},[586,3900,3901,3915],{},[589,3902,3903,3905,3907,3909,3911,3913],{},[611,3904,848],{},[596,3906,852],{"stretchy":851},[592,3908,1759],{},[596,3910,857],{"stretchy":851},[596,3912,860],{},[592,3914,1946],{},[615,3916,3917],{"encoding":617},"\\varphi(12) = 4",[573,3919,3921,3948],{"className":3920,"ariaHidden":598},[622],[573,3922,3924,3927,3930,3933,3936,3939,3942,3945],{"className":3923},[626],[573,3925],{"className":3926,"style":880},[630],[573,3928,848],{"className":3929},[635,670],[573,3931,852],{"className":3932},[887],[573,3934,1759],{"className":3935},[635],[573,3937,857],{"className":3938},[894],[573,3940],{"className":3941,"style":898},[643],[573,3943,860],{"className":3944},[902],[573,3946],{"className":3947,"style":898},[643],[573,3949,3951,3954],{"className":3950},[626],[573,3952],{"className":3953,"style":760},[630],[573,3955,1946],{"className":3956},[635],", nicht ",[573,3959,3961,3974],{"className":3960},[576],[573,3962,3964],{"className":3963},[580],[582,3965,3966],{"xmlns":584},[586,3967,3968,3972],{},[589,3969,3970],{},[592,3971,2275],{},[615,3973,2275],{"encoding":617},[573,3975,3977],{"className":3976,"ariaHidden":598},[622],[573,3978,3980,3983],{"className":3979},[626],[573,3981],{"className":3982,"style":760},[630],[573,3984,2275],{"className":3985},[635],{"title":3987,"searchDepth":3988,"depth":3988,"links":3989},"",2,[3990,3991,3992],{"id":562,"depth":3988,"text":563},{"id":1737,"depth":3988,"text":1738},{"id":3293,"depth":3988,"text":3294},[3994,3996,3998,4000,4002,4004,4006,4008,4011,4013,4015],{"lhs":3995,"rhs":1946},"12*(1 - 1\u002F2)*(1 - 1\u002F3)",{"lhs":3997,"rhs":1946},"sum(map(1:12, f(k) = gcd(k, 12) == 1 ? 1 : 0))",{"lhs":3999,"rhs":2520},"100*(1 - 1\u002F2)*(1 - 1\u002F5)",{"lhs":4001,"rhs":2520},"sum(map(1:100, f(k) = gcd(k, 100) == 1 ? 1 : 0))",{"lhs":4003,"rhs":2873},"3^4 - 3^3",{"lhs":4005,"rhs":2873},"81*(1 - 1\u002F3)",{"lhs":4007,"rhs":1759},"13 - 1",{"lhs":4009,"rhs":4010},"36*(1 - 1\u002F2)*(1 - 1\u002F3)","(4 - 2)*(9 - 3)",{"lhs":4012,"rhs":1759},"sum(map(1:36, f(k) = gcd(k, 36) == 1 ? 1 : 0))",{"lhs":4014,"rhs":602},"sum(map(1:4, f(k) = gcd(k, 4) == 1 ? 1 : 0))",{"lhs":4016,"rhs":594},"(2 - 1)*(2 - 1)","n \\in \\mathbb{N},\\ n \\geq 1;\\ \\text{Produkt über die verschiedenen Primteiler } p \\text{ von } n","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">N\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n \\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\">n\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>n\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n \\geq 1;\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\">n\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:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">;\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Produkt über die verschiedenen Primteiler\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">von\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","euler-totient","\\varphi(n) = n \\prod_{p \\mid n} \\left(1 - \\frac{1}{p}\\right)","\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>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmunder>\u003Cmo>∏\u003C\u002Fmo>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi(n) = n \\prod_{p \\mid n} \\left(1 - \\frac{1}{p}\\right)\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:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\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:2.966em;vertical-align:-1.516em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.05em;\">\u003Cspan style=\"top:-1.809em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∏\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.516em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8804em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Feulersche-phi-funktion",[],[4030,4031,4032,4033],"euler-theorem","prime-factorization","gcd-lcm","chinese-remainder",{"title":548,"description":3987},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Feulersche-phi-funktion","φ(n) zählt die zu n teilerfremden Zahlen von 1 bis n und ergibt sich aus den verschiedenen Primteilern von n.",[4038,704,4039,4040],"phi-funktion","primfaktoren","zahlentheorie",[4042,4046,4049],{"symbol":4043,"meaning":4044,"symbolHtml":4045},"\\varphi(n)","Anzahl der Zahlen k mit 1 ≤ k ≤ n und ggT(k, n) = 1","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi(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:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":613,"meaning":4047,"symbolHtml":4048},"natürliche Zahl ab 1","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":565,"meaning":4050,"symbolHtml":4051},"die verschiedenen Primzahlen, die n teilen (jede nur einmal)","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","jaUMpv1jpzb1nVr6cOIJGMYUTNW1ZmcIXzimhTvdgKY",[4054,4057,4060,4063],{"stem":4055,"title":4056,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fggt-und-kgv","Zusammenhang von ggT und kgV",{"stem":4058,"title":4059,"category":65},"de\u002Fformulas\u002Fmath\u002Farithmetic\u002Fprimfaktorzerlegung","Primfaktorzerlegung",{"stem":4061,"title":4062,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fchinesischer-restsatz","Chinesischer Restsatz",{"stem":4064,"title":4065,"category":187},"de\u002Fformulas\u002Fmath\u002Fdiscrete-math\u002Fsatz-von-euler","Satz von Euler",1793824276055]