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werden durch Multiplikation mit ",[573,1240,1242,1255],{"className":1241},[576],[573,1243,1245],{"className":1244},[580],[582,1246,1247],{"xmlns":584},[586,1248,1249,1253],{},[589,1250,1251],{},[592,1252,633],{},[596,1254,633],{"encoding":598},[573,1256,1258],{"className":1257,"ariaHidden":603},[602],[573,1259,1261,1264],{"className":1260},[607],[573,1262],{"className":1263,"style":612},[611],[573,1265,633],{"className":1266},[616,617]," nur umgeordnet. Vergleicht man die Produkte, folgt 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und das Produkt darf man kürzen, weil es zu ",[573,1566,1568,1581],{"className":1567},[576],[573,1569,1571],{"className":1570},[580],[582,1572,1573],{"xmlns":584},[586,1574,1575,1579],{},[589,1576,1577],{},[592,1578,594],{},[596,1580,594],{"encoding":598},[573,1582,1584],{"className":1583,"ariaHidden":603},[602],[573,1585,1587,1590],{"className":1586},[607],[573,1588],{"className":1589,"style":612},[611],[573,1591,594],{"className":1592},[616,617]," teilerfremd ist.",[565,1595,1596],{},"Folgerungen:",[1598,1599,1600,1920,2072],"ul",{},[1601,1602,1603,1606,1607,1832,1833,1919],"li",{},[569,1604,1605],{},"Exponenten reduzieren:"," 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",[573,2079,2081,2105],{"className":2080},[576],[573,2082,2084],{"className":2083},[580],[582,2085,2086],{"xmlns":584},[586,2087,2088,2102],{},[589,2089,2090,2093,2095,2097,2099],{},[592,2091,2092],{},"n",[701,2094,853],{},[592,2096,565],{},[701,2098,1296],{},[592,2100,2101],{},"q",[596,2103,2104],{"encoding":598},"n = p \\cdot q",[573,2106,2108,2126,2145],{"className":2107,"ariaHidden":603},[602],[573,2109,2111,2114,2117,2120,2123],{"className":2110},[607],[573,2112],{"className":2113,"style":612},[611],[573,2115,2092],{"className":2116},[616,617],[573,2118],{"className":2119,"style":874},[788],[573,2121,853],{"className":2122},[878],[573,2124],{"className":2125,"style":874},[788],[573,2127,2129,2133,2136,2139,2142],{"className":2128},[607],[573,2130],{"className":2131,"style":2132},[611],"height:0.6389em;vertical-align:-0.1944em;",[573,2134,565],{"className":2135},[616,617],[573,2137],{"className":2138,"style":789},[788],[573,2140,1296],{"className":2141},[793],[573,2143],{"className":2144,"style":789},[788],[573,2146,2148,2151],{"className":2147},[607],[573,2149],{"className":2150,"style":888},[611],[573,2152,2101],{"className":2153,"style":2154},[616,617],"margin-right:0.0359em;"," mit verschiedenen Primzahlen ",[573,2157,2159,2177],{"className":2158},[576],[573,2160,2162],{"className":2161},[580],[582,2163,2164],{"xmlns":584},[586,2165,2166,2174],{},[589,2167,2168,2170,2172],{},[592,2169,565],{},[701,2171,1082],{"separator":603},[592,2173,2101],{},[596,2175,2176],{"encoding":598},"p, q",[573,2178,2180],{"className":2179,"ariaHidden":603},[602],[573,2181,2183,2186,2189,2192,2195],{"className":2182},[607],[573,2184],{"className":2185,"style":888},[611],[573,2187,565],{"className":2188},[616,617],[573,2190,1082],{"className":2191},[1165],[573,2193],{"className":2194,"style":1169},[788],[573,2196,2101],{"className":2197,"style":2154},[616,617]," und ",[573,2200,2202,2251],{"className":2201},[576],[573,2203,2205],{"className":2204},[580],[582,2206,2207],{"xmlns":584},[586,2208,2209,2248],{},[589,2210,2211,2213,2215,2218,2220,2222,2224,2226,2228,2236,2238,2240,2242,2244,2246],{},[592,2212,1335],{},[701,2214,1296],{},[592,2216,2217],{},"e",[701,2219,1309],{},[714,2221,716],{},[788,2223],{},[788,2225],{"width":1322},[701,2227,704],{"stretchy":703},[589,2229,2230,2232,2234],{},[592,2231,594],{"mathvariant":1329},[592,2233,1332],{"mathvariant":1329},[592,2235,1335],{"mathvariant":1329},[788,2237],{"width":1338},[592,2239,699],{},[701,2241,704],{"stretchy":703},[592,2243,2092],{},[701,2245,709],{"stretchy":703},[701,2247,709],{"stretchy":703},[596,2249,2250],{"encoding":598},"d \\cdot e \\equiv 1 \\pmod{\\varphi(n)}",[573,2252,2254,2273,2292,2307],{"className":2253,"ariaHidden":603},[602],[573,2255,2257,2261,2264,2267,2270],{"className":2256},[607],[573,2258],{"className":2259,"style":2260},[611],"height:0.6944em;",[573,2262,1335],{"className":2263},[616,617],[573,2265],{"className":2266,"style":789},[788],[573,2268,1296],{"className":2269},[793],[573,2271],{"className":2272,"style":789},[788],[573,2274,2276,2280,2283,2286,2289],{"className":2275},[607],[573,2277],{"className":2278,"style":2279},[611],"height:0.4637em;",[573,2281,2217],{"className":2282},[616,617],[573,2284],{"className":2285,"style":874},[788],[573,2287,1309],{"className":2288},[878],[573,2290],{"className":2291,"style":874},[788],[573,2293,2295,2298,2301,2304],{"className":2294},[607],[573,2296],{"className":2297,"style":803},[611],[573,2299,716],{"className":2300},[616],[573,2302],{"className":2303},[788,1529],[573,2305],{"className":2306,"style":1533},[788],[573,2308,2310,2313,2316,2325,2328,2331,2334,2337],{"className":2309},[607],[573,2311],{"className":2312,"style":933},[611],[573,2314,704],{"className":2315},[777],[573,2317,2319],{"className":2318},[616],[573,2320,2322],{"className":2321},[616],[573,2323,1553],{"className":2324},[616,1552],[573,2326],{"className":2327,"style":1557},[788],[573,2329,699],{"className":2330},[616,617],[573,2332,704],{"className":2333},[777],[573,2335,2092],{"className":2336},[616,617],[573,2338,2340],{"className":2339},[784],"))"," gilt ",[573,2343,2345,2396],{"className":2344},[576],[573,2346,2348],{"className":2347},[580],[582,2349,2350],{"xmlns":584},[586,2351,2352,2393],{},[589,2353,2354,2356,2363,2369,2371,2373,2375,2377,2379,2387,2389,2391],{},[701,2355,704],{"stretchy":703},[691,2357,2358,2361],{},[592,2359,2360],{},"x",[592,2362,2217],{},[691,2364,2365,2367],{},[701,2366,709],{"stretchy":703},[592,2368,1335],{},[701,2370,1309],{},[592,2372,2360],{},[788,2374],{},[788,2376],{"width":1322},[701,2378,704],{"stretchy":703},[589,2380,2381,2383,2385],{},[592,2382,594],{"mathvariant":1329},[592,2384,1332],{"mathvariant":1329},[592,2386,1335],{"mathvariant":1329},[788,2388],{"width":1338},[592,2390,2092],{},[701,2392,709],{"stretchy":703},[596,2394,2395],{"encoding":598},"(x^e)^d \\equiv x \\pmod n",[573,2397,2399,2477,2492],{"className":2398,"ariaHidden":603},[602],[573,2400,2402,2406,2409,2439,2468,2471,2474],{"className":2401},[607],[573,2403],{"className":2404,"style":2405},[611],"height:1.0991em;vertical-align:-0.25em;",[573,2407,704],{"className":2408},[777],[573,2410,2412,2415],{"className":2411},[616],[573,2413,2360],{"className":2414},[616,617],[573,2416,2418],{"className":2417},[739],[573,2419,2421],{"className":2420},[743],[573,2422,2424],{"className":2423},[747],[573,2425,2428],{"className":2426,"style":2427},[751],"height:0.6644em;",[573,2429,2430,2433],{"style":755},[573,2431],{"className":2432,"style":760},[759],[573,2434,2436],{"className":2435},[764,765,766,767],[573,2437,2217],{"className":2438},[616,617,767],[573,2440,2442,2445],{"className":2441},[784],[573,2443,709],{"className":2444},[784],[573,2446,2448],{"className":2447},[739],[573,2449,2451],{"className":2450},[743],[573,2452,2454],{"className":2453},[747],[573,2455,2457],{"className":2456,"style":1693},[751],[573,2458,2459,2462],{"style":755},[573,2460],{"className":2461,"style":760},[759],[573,2463,2465],{"className":2464},[764,765,766,767],[573,2466,1335],{"className":2467},[616,617,767],[573,2469],{"className":2470,"style":874},[788],[573,2472,1309],{"className":2473},[878],[573,2475],{"className":2476,"style":874},[788],[573,2478,2480,2483,2486,2489],{"className":2479},[607],[573,2481],{"className":2482,"style":612},[611],[573,2484,2360],{"className":2485},[616,617],[573,2487],{"className":2488},[788,1529],[573,2490],{"className":2491,"style":1533},[788],[573,2493,2495,2498,2501,2510,2513,2516],{"className":2494},[607],[573,2496],{"className":2497,"style":933},[611],[573,2499,704],{"className":2500},[777],[573,2502,2504],{"className":2503},[616],[573,2505,2507],{"className":2506},[616],[573,2508,1553],{"className":2509},[616,1552],[573,2511],{"className":2512,"style":1557},[788],[573,2514,2092],{"className":2515},[616,617],[573,2517,709],{"className":2518},[784],"; darauf beruhen Ver- und Entschlüsselung.",[560,2521,2523],{"id":2522},"beispiele","Beispiele",[565,2525,2526,2595,2596,2625,2626,2198,2693,2777,2778,2875],{},[569,2527,2528,2529,2594],{},"Letzte zwei Ziffern von ",[573,2530,2532,2552],{"className":2531},[576],[573,2533,2535],{"className":2534},[580],[582,2536,2537],{"xmlns":584},[586,2538,2539,2549],{},[589,2540,2541],{},[691,2542,2543,2546],{},[714,2544,2545],{},"3",[714,2547,2548],{},"123",[596,2550,2551],{"encoding":598},"3^{123}",[573,2553,2555],{"className":2554,"ariaHidden":603},[602],[573,2556,2558,2562],{"className":2557},[607],[573,2559],{"className":2560,"style":2561},[611],"height:0.8141em;",[573,2563,2565,2568],{"className":2564},[616],[573,2566,2545],{"className":2567},[616],[573,2569,2571],{"className":2570},[739],[573,2572,2574],{"className":2573},[743],[573,2575,2577],{"className":2576},[747],[573,2578,2580],{"className":2579,"style":2561},[751],[573,2581,2582,2585],{"style":755},[573,2583],{"className":2584,"style":760},[759],[573,2586,2588],{"className":2587},[764,765,766,767],[573,2589,2591],{"className":2590},[616,767],[573,2592,2548],{"className":2593},[616,767],":"," Gesucht ist der Rest modulo ",[573,2597,2599,2613],{"className":2598},[576],[573,2600,2602],{"className":2601},[580],[582,2603,2604],{"xmlns":584},[586,2605,2606,2611],{},[589,2607,2608],{},[714,2609,2610],{},"100",[596,2612,2610],{"encoding":598},[573,2614,2616],{"className":2615,"ariaHidden":603},[602],[573,2617,2619,2622],{"className":2618},[607],[573,2620],{"className":2621,"style":803},[611],[573,2623,2610],{"className":2624},[616],". Es gilt ",[573,2627,2629,2654],{"className":2628},[576],[573,2630,2632],{"className":2631},[580],[582,2633,2634],{"xmlns":584},[586,2635,2636,2651],{},[589,2637,2638,2640,2642,2644,2646,2648],{},[592,2639,699],{},[701,2641,704],{"stretchy":703},[714,2643,2610],{},[701,2645,709],{"stretchy":703},[701,2647,853],{},[714,2649,2650],{},"40",[596,2652,2653],{"encoding":598},"\\varphi(100) = 40",[573,2655,2657,2684],{"className":2656,"ariaHidden":603},[602],[573,2658,2660,2663,2666,2669,2672,2675,2678,2681],{"className":2659},[607],[573,2661],{"className":2662,"style":933},[611],[573,2664,699],{"className":2665},[616,617],[573,2667,704],{"className":2668},[777],[573,2670,2610],{"className":2671},[616],[573,2673,709],{"className":2674},[784],[573,2676],{"className":2677,"style":874},[788],[573,2679,853],{"className":2680},[878],[573,2682],{"className":2683,"style":874},[788],[573,2685,2687,2690],{"className":2686},[607],[573,2688],{"className":2689,"style":803},[611],[573,2691,2650],{"className":2692},[616],[573,2694,2696,2726],{"className":2695},[576],[573,2697,2699],{"className":2698},[580],[582,2700,2701],{"xmlns":584},[586,2702,2703,2723],{},[589,2704,2705,2707,2709,2711,2713,2715,2717,2719,2721],{},[592,2706,1847],{"mathvariant":1329},[701,2708,1850],{},[701,2710,704],{"stretchy":703},[714,2712,2545],{},[701,2714,1082],{"separator":603},[714,2716,2610],{},[701,2718,709],{"stretchy":703},[701,2720,853],{},[714,2722,716],{},[596,2724,2725],{"encoding":598},"\\operatorname{ggT}(3, 100) = 1",[573,2727,2729,2768],{"className":2728,"ariaHidden":603},[602],[573,2730,2732,2735,2741,2744,2747,2750,2753,2756,2759,2762,2765],{"className":2731},[607],[573,2733],{"className":2734,"style":933},[611],[573,2736,2738],{"className":2737},[1415],[573,2739,1847],{"className":2740},[616,1552],[573,2742,704],{"className":2743},[777],[573,2745,2545],{"className":2746},[616],[573,2748,1082],{"className":2749},[1165],[573,2751],{"className":2752,"style":1169},[788],[573,2754,2610],{"className":2755},[616],[573,2757,709],{"className":2758},[784],[573,2760],{"className":2761,"style":874},[788],[573,2763,853],{"className":2764},[878],[573,2766],{"className":2767,"style":874},[788],[573,2769,2771,2774],{"className":2770},[607],[573,2772],{"className":2773,"style":803},[611],[573,2775,716],{"className":2776},[616],". Mit ",[573,2779,2781,2808],{"className":2780},[576],[573,2782,2784],{"className":2783},[580],[582,2785,2786],{"xmlns":584},[586,2787,2788,2805],{},[589,2789,2790,2792,2794,2796,2798,2800,2803],{},[714,2791,2548],{},[701,2793,853],{},[714,2795,2545],{},[701,2797,1296],{},[714,2799,2650],{},[701,2801,2802],{},"+",[714,2804,2545],{},[596,2806,2807],{"encoding":598},"123 = 3 \\cdot 40 + 3",[573,2809,2811,2829,2847,2866],{"className":2810,"ariaHidden":603},[602],[573,2812,2814,2817,2820,2823,2826],{"className":2813},[607],[573,2815],{"className":2816,"style":803},[611],[573,2818,2548],{"className":2819},[616],[573,2821],{"className":2822,"style":874},[788],[573,2824,853],{"className":2825},[878],[573,2827],{"className":2828,"style":874},[788],[573,2830,2832,2835,2838,2841,2844],{"className":2831},[607],[573,2833],{"className":2834,"style":803},[611],[573,2836,2545],{"className":2837},[616],[573,2839],{"className":2840,"style":789},[788],[573,2842,1296],{"className":2843},[793],[573,2845],{"className":2846,"style":789},[788],[573,2848,2850,2854,2857,2860,2863],{"className":2849},[607],[573,2851],{"className":2852,"style":2853},[611],"height:0.7278em;vertical-align:-0.0833em;",[573,2855,2650],{"className":2856},[616],[573,2858],{"className":2859,"style":789},[788],[573,2861,2802],{"className":2862},[793],[573,2864],{"className":2865,"style":789},[788],[573,2867,2869,2872],{"className":2868},[607],[573,2870],{"className":2871,"style":803},[611],[573,2873,2545],{"className":2874},[616]," folgt",[573,2877,2880],{"className":2878},[2879],"katex-display",[573,2881,2883,2962],{"className":2882},[576],[573,2884,2886],{"className":2885},[580],[582,2887,2889],{"xmlns":584,"display":2888},"block",[586,2890,2891,2959],{},[589,2892,2893,2899,2901,2917,2919,2925,2927,2929,2931,2934,2936,2938,2940,2943,2945,2953,2955,2957],{},[691,2894,2895,2897],{},[714,2896,2545],{},[714,2898,2548],{},[701,2900,853],{},[691,2902,2903,2915],{},[589,2904,2905,2907,2913],{},[701,2906,704],{"fence":603},[691,2908,2909,2911],{},[714,2910,2545],{},[714,2912,2650],{},[701,2914,709],{"fence":603},[714,2916,2545],{},[701,2918,1296],{},[691,2920,2921,2923],{},[714,2922,2545],{},[714,2924,2545],{},[701,2926,1309],{},[714,2928,716],{},[701,2930,1296],{},[714,2932,2933],{},"27",[701,2935,853],{},[714,2937,2933],{},[788,2939],{},[788,2941],{"width":2942},"1em",[701,2944,704],{"stretchy":703},[589,2946,2947,2949,2951],{},[592,2948,594],{"mathvariant":1329},[592,2950,1332],{"mathvariant":1329},[592,2952,1335],{"mathvariant":1329},[788,2954],{"width":1338},[714,2956,2610],{},[701,2958,709],{"stretchy":703},[596,2960,2961],{"encoding":598},"3^{123} = \\left(3^{40}\\right)^3 \\cdot 3^3 \\equiv 1 \\cdot 27 = 27 \\pmod{100}",[573,2963,2965,3014,3109,3153,3171,3189,3205],{"className":2964,"ariaHidden":603},[602],[573,2966,2968,2972,3005,3008,3011],{"className":2967},[607],[573,2969],{"className":2970,"style":2971},[611],"height:0.8641em;",[573,2973,2975,2978],{"className":2974},[616],[573,2976,2545],{"className":2977},[616],[573,2979,2981],{"className":2980},[739],[573,2982,2984],{"className":2983},[743],[573,2985,2987],{"className":2986},[747],[573,2988,2990],{"className":2989,"style":2971},[751],[573,2991,2993,2996],{"style":2992},"top:-3.113em;margin-right:0.05em;",[573,2994],{"className":2995,"style":760},[759],[573,2997,2999],{"className":2998},[764,765,766,767],[573,3000,3002],{"className":3001},[616,767],[573,3003,2548],{"className":3004},[616,767],[573,3006],{"className":3007,"style":874},[788],[573,3009,853],{"className":3010},[878],[573,3012],{"className":3013,"style":874},[788],[573,3015,3017,3021,3100,3103,3106],{"className":3016},[607],[573,3018],{"className":3019,"style":3020},[611],"height:1.4181em;vertical-align:-0.35em;",[573,3022,3024,3075],{"className":3023},[1173],[573,3025,3027,3037,3069],{"className":3026},[1173],[573,3028,3032],{"className":3029,"style":3031},[777,3030],"delimcenter","top:0em;",[573,3033,704],{"className":3034},[3035,3036],"delimsizing","size1",[573,3038,3040,3043],{"className":3039},[616],[573,3041,2545],{"className":3042},[616],[573,3044,3046],{"className":3045},[739],[573,3047,3049],{"className":3048},[743],[573,3050,3052],{"className":3051},[747],[573,3053,3055],{"className":3054,"style":2971},[751],[573,3056,3057,3060],{"style":2992},[573,3058],{"className":3059,"style":760},[759],[573,3061,3063],{"className":3062},[764,765,766,767],[573,3064,3066],{"className":3065},[616,767],[573,3067,2650],{"className":3068},[616,767],[573,3070,3072],{"className":3071,"style":3031},[784,3030],[573,3073,709],{"className":3074},[3035,3036],[573,3076,3078],{"className":3077},[739],[573,3079,3081],{"className":3080},[743],[573,3082,3084],{"className":3083},[747],[573,3085,3088],{"className":3086,"style":3087},[751],"height:1.0681em;",[573,3089,3091,3094],{"style":3090},"top:-3.317em;margin-right:0.05em;",[573,3092],{"className":3093,"style":760},[759],[573,3095,3097],{"className":3096},[764,765,766,767],[573,3098,2545],{"className":3099},[616,767],[573,3101],{"className":3102,"style":789},[788],[573,3104,1296],{"className":3105},[793],[573,3107],{"className":3108,"style":789},[788],[573,3110,3112,3115,3144,3147,3150],{"className":3111},[607],[573,3113],{"className":3114,"style":2971},[611],[573,3116,3118,3121],{"className":3117},[616],[573,3119,2545],{"className":3120},[616],[573,3122,3124],{"className":3123},[739],[573,3125,3127],{"className":3126},[743],[573,3128,3130],{"className":3129},[747],[573,3131,3133],{"className":3132,"style":2971},[751],[573,3134,3135,3138],{"style":2992},[573,3136],{"className":3137,"style":760},[759],[573,3139,3141],{"className":3140},[764,765,766,767],[573,3142,2545],{"className":3143},[616,767],[573,3145],{"className":3146,"style":874},[788],[573,3148,1309],{"className":3149},[878],[573,3151],{"className":3152,"style":874},[788],[573,3154,3156,3159,3162,3165,3168],{"className":3155},[607],[573,3157],{"className":3158,"style":803},[611],[573,3160,716],{"className":3161},[616],[573,3163],{"className":3164,"style":789},[788],[573,3166,1296],{"className":3167},[793],[573,3169],{"className":3170,"style":789},[788],[573,3172,3174,3177,3180,3183,3186],{"className":3173},[607],[573,3175],{"className":3176,"style":803},[611],[573,3178,2933],{"className":3179},[616],[573,3181],{"className":3182,"style":874},[788],[573,3184,853],{"className":3185},[878],[573,3187],{"className":3188,"style":874},[788],[573,3190,3192,3195,3198,3201],{"className":3191},[607],[573,3193],{"className":3194,"style":803},[611],[573,3196,2933],{"className":3197},[616],[573,3199],{"className":3200},[788,1529],[573,3202],{"className":3203,"style":3204},[788],"margin-right:1em;",[573,3206,3208,3211,3214,3223,3226,3229],{"className":3207},[607],[573,3209],{"className":3210,"style":933},[611],[573,3212,704],{"className":3213},[777],[573,3215,3217],{"className":3216},[616],[573,3218,3220],{"className":3219},[616],[573,3221,1553],{"className":3222},[616,1552],[573,3224],{"className":3225,"style":1557},[788],[573,3227,2610],{"className":3228},[616],[573,3230,709],{"className":3231},[784],[565,3233,3234,3235,1919],{},"Die letzten beiden Ziffern sind ",[573,3236,3238,3251],{"className":3237},[576],[573,3239,3241],{"className":3240},[580],[582,3242,3243],{"xmlns":584},[586,3244,3245,3249],{},[589,3246,3247],{},[714,3248,2933],{},[596,3250,2933],{"encoding":598},[573,3252,3254],{"className":3253,"ariaHidden":603},[602],[573,3255,3257,3260],{"className":3256},[607],[573,3258],{"className":3259,"style":803},[611],[573,3261,2933],{"className":3262},[616],[565,3264,3265,1606,3325,3392,3393,3565,3566,1919],{},[569,3266,3267,3268,2043,3296,2594],{},"Inverses von ",[573,3269,3271,3284],{"className":3270},[576],[573,3272,3274],{"className":3273},[580],[582,3275,3276],{"xmlns":584},[586,3277,3278,3282],{},[589,3279,3280],{},[714,3281,2545],{},[596,3283,2545],{"encoding":598},[573,3285,3287],{"className":3286,"ariaHidden":603},[602],[573,3288,3290,3293],{"className":3289},[607],[573,3291],{"className":3292,"style":803},[611],[573,3294,2545],{"className":3295},[616],[573,3297,3299,3313],{"className":3298},[576],[573,3300,3302],{"className":3301},[580],[582,3303,3304],{"xmlns":584},[586,3305,3306,3311],{},[589,3307,3308],{},[714,3309,3310],{},"10",[596,3312,3310],{"encoding":598},[573,3314,3316],{"className":3315,"ariaHidden":603},[602],[573,3317,3319,3322],{"className":3318},[607],[573,3320],{"className":3321,"style":803},[611],[573,3323,3310],{"className":3324},[616],[573,3326,3328,3353],{"className":3327},[576],[573,3329,3331],{"className":3330},[580],[582,3332,3333],{"xmlns":584},[586,3334,3335,3350],{},[589,3336,3337,3339,3341,3343,3345,3347],{},[592,3338,699],{},[701,3340,704],{"stretchy":703},[714,3342,3310],{},[701,3344,709],{"stretchy":703},[701,3346,853],{},[714,3348,3349],{},"4",[596,3351,3352],{"encoding":598},"\\varphi(10) = 4",[573,3354,3356,3383],{"className":3355,"ariaHidden":603},[602],[573,3357,3359,3362,3365,3368,3371,3374,3377,3380],{"className":3358},[607],[573,3360],{"className":3361,"style":933},[611],[573,3363,699],{"className":3364},[616,617],[573,3366,704],{"className":3367},[777],[573,3369,3310],{"className":3370},[616],[573,3372,709],{"className":3373},[784],[573,3375],{"className":3376,"style":874},[788],[573,3378,853],{"className":3379},[878],[573,3381],{"className":3382,"style":874},[788],[573,3384,3386,3389],{"className":3385},[607],[573,3387],{"className":3388,"style":803},[611],[573,3390,3349],{"className":3391},[616],", also ist ",[573,3394,3396,3449],{"className":3395},[576],[573,3397,3399],{"className":3398},[580],[582,3400,3401],{"xmlns":584},[586,3402,3403,3446],{},[589,3404,3405,3417,3419,3421,3423,3426,3428,3430,3432,3440,3442,3444],{},[691,3406,3407,3409],{},[714,3408,2545],{},[589,3410,3411,3413,3415],{},[714,3412,3349],{},[701,3414,712],{},[714,3416,716],{},[701,3418,853],{},[714,3420,2933],{},[701,3422,1309],{},[714,3424,3425],{},"7",[788,3427],{},[788,3429],{"width":1322},[701,3431,704],{"stretchy":703},[589,3433,3434,3436,3438],{},[592,3435,594],{"mathvariant":1329},[592,3437,1332],{"mathvariant":1329},[592,3439,1335],{"mathvariant":1329},[788,3441],{"width":1338},[714,3443,3310],{},[701,3445,709],{"stretchy":703},[596,3447,3448],{"encoding":598},"3^{4-1} = 27 \\equiv 7 \\pmod{10}",[573,3450,3452,3505,3523,3538],{"className":3451,"ariaHidden":603},[602],[573,3453,3455,3458,3496,3499,3502],{"className":3454},[607],[573,3456],{"className":3457,"style":2561},[611],[573,3459,3461,3464],{"className":3460},[616],[573,3462,2545],{"className":3463},[616],[573,3465,3467],{"className":3466},[739],[573,3468,3470],{"className":3469},[743],[573,3471,3473],{"className":3472},[747],[573,3474,3476],{"className":3475,"style":2561},[751],[573,3477,3478,3481],{"style":755},[573,3479],{"className":3480,"style":760},[759],[573,3482,3484],{"className":3483},[764,765,766,767],[573,3485,3487,3490,3493],{"className":3486},[616,767],[573,3488,3349],{"className":3489},[616,767],[573,3491,712],{"className":3492},[793,767],[573,3494,716],{"className":3495},[616,767],[573,3497],{"className":3498,"style":874},[788],[573,3500,853],{"className":3501},[878],[573,3503],{"className":3504,"style":874},[788],[573,3506,3508,3511,3514,3517,3520],{"className":3507},[607],[573,3509],{"className":3510,"style":803},[611],[573,3512,2933],{"className":3513},[616],[573,3515],{"className":3516,"style":874},[788],[573,3518,1309],{"className":3519},[878],[573,3521],{"className":3522,"style":874},[788],[573,3524,3526,3529,3532,3535],{"className":3525},[607],[573,3527],{"className":3528,"style":803},[611],[573,3530,3425],{"className":3531},[616],[573,3533],{"className":3534},[788,1529],[573,3536],{"className":3537,"style":1533},[788],[573,3539,3541,3544,3547,3556,3559,3562],{"className":3540},[607],[573,3542],{"className":3543,"style":933},[611],[573,3545,704],{"className":3546},[777],[573,3548,3550],{"className":3549},[616],[573,3551,3553],{"className":3552},[616],[573,3554,1553],{"className":3555},[616,1552],[573,3557],{"className":3558,"style":1557},[788],[573,3560,3310],{"className":3561},[616],[573,3563,709],{"className":3564},[784]," das Inverse. Probe: ",[573,3567,3569,3616],{"className":3568},[576],[573,3570,3572],{"className":3571},[580],[582,3573,3574],{"xmlns":584},[586,3575,3576,3613],{},[589,3577,3578,3580,3582,3584,3586,3589,3591,3593,3595,3597,3599,3607,3609,3611],{},[714,3579,2545],{},[701,3581,1296],{},[714,3583,3425],{},[701,3585,853],{},[714,3587,3588],{},"21",[701,3590,1309],{},[714,3592,716],{},[788,3594],{},[788,3596],{"width":1322},[701,3598,704],{"stretchy":703},[589,3600,3601,3603,3605],{},[592,3602,594],{"mathvariant":1329},[592,3604,1332],{"mathvariant":1329},[592,3606,1335],{"mathvariant":1329},[788,3608],{"width":1338},[714,3610,3310],{},[701,3612,709],{"stretchy":703},[596,3614,3615],{"encoding":598},"3 \\cdot 7 = 21 \\equiv 1 \\pmod{10}",[573,3617,3619,3637,3655,3673,3688],{"className":3618,"ariaHidden":603},[602],[573,3620,3622,3625,3628,3631,3634],{"className":3621},[607],[573,3623],{"className":3624,"style":803},[611],[573,3626,2545],{"className":3627},[616],[573,3629],{"className":3630,"style":789},[788],[573,3632,1296],{"className":3633},[793],[573,3635],{"className":3636,"style":789},[788],[573,3638,3640,3643,3646,3649,3652],{"className":3639},[607],[573,3641],{"className":3642,"style":803},[611],[573,3644,3425],{"className":3645},[616],[573,3647],{"className":3648,"style":874},[788],[573,3650,853],{"className":3651},[878],[573,3653],{"className":3654,"style":874},[788],[573,3656,3658,3661,3664,3667,3670],{"className":3657},[607],[573,3659],{"className":3660,"style":803},[611],[573,3662,3588],{"className":3663},[616],[573,3665],{"className":3666,"style":874},[788],[573,3668,1309],{"className":3669},[878],[573,3671],{"className":3672,"style":874},[788],[573,3674,3676,3679,3682,3685],{"className":3675},[607],[573,3677],{"className":3678,"style":803},[611],[573,3680,716],{"className":3681},[616],[573,3683],{"className":3684},[788,1529],[573,3686],{"className":3687,"style":1533},[788],[573,3689,3691,3694,3697,3706,3709,3712],{"className":3690},[607],[573,3692],{"className":3693,"style":933},[611],[573,3695,704],{"className":3696},[777],[573,3698,3700],{"className":3699},[616],[573,3701,3703],{"className":3702},[616],[573,3704,1553],{"className":3705},[616,1552],[573,3707],{"className":3708,"style":1557},[788],[573,3710,3310],{"className":3711},[616],[573,3713,709],{"className":3714},[784],[560,3716,3718],{"id":3717},"typische-fehler","Typische Fehler",[1598,3720,3721,4089,4293],{},[1601,3722,3723,2077,3726,3778,3779,892,3830,4060,4061,1919],{},[569,3724,3725],{},"Die Teilerfremdheit nicht prüfen.",[573,3727,3729,3748],{"className":3728},[576],[573,3730,3732],{"className":3731},[580],[582,3733,3734],{"xmlns":584},[586,3735,3736,3745],{},[589,3737,3738,3740,3742],{},[592,3739,633],{},[701,3741,853],{},[714,3743,3744],{},"2",[596,3746,3747],{"encoding":598},"a = 2",[573,3749,3751,3769],{"className":3750,"ariaHidden":603},[602],[573,3752,3754,3757,3760,3763,3766],{"className":3753},[607],[573,3755],{"className":3756,"style":612},[611],[573,3758,633],{"className":3759},[616,617],[573,3761],{"className":3762,"style":874},[788],[573,3764,853],{"className":3765},[878],[573,3767],{"className":3768,"style":874},[788],[573,3770,3772,3775],{"className":3771},[607],[573,3773],{"className":3774,"style":803},[611],[573,3776,3744],{"className":3777},[616],", ",[573,3780,3782,3800],{"className":3781},[576],[573,3783,3785],{"className":3784},[580],[582,3786,3787],{"xmlns":584},[586,3788,3789,3797],{},[589,3790,3791,3793,3795],{},[592,3792,594],{},[701,3794,853],{},[714,3796,3310],{},[596,3798,3799],{"encoding":598},"m = 10",[573,3801,3803,3821],{"className":3802,"ariaHidden":603},[602],[573,3804,3806,3809,3812,3815,3818],{"className":3805},[607],[573,3807],{"className":3808,"style":612},[611],[573,3810,594],{"className":3811},[616,617],[573,3813],{"className":3814,"style":874},[788],[573,3816,853],{"className":3817},[878],[573,3819],{"className":3820,"style":874},[788],[573,3822,3824,3827],{"className":3823},[607],[573,3825],{"className":3826,"style":803},[611],[573,3828,3310],{"className":3829},[616],[573,3831,3833,3897],{"className":3832},[576],[573,3834,3836],{"className":3835},[580],[582,3837,3838],{"xmlns":584},[586,3839,3840,3894],{},[589,3841,3842,3856,3858,3864,3866,3869,3871,3874,3876,3878,3880,3888,3890,3892],{},[691,3843,3844,3846],{},[714,3845,3744],{},[589,3847,3848,3850,3852,3854],{},[592,3849,699],{},[701,3851,704],{"stretchy":703},[714,3853,3310],{},[701,3855,709],{"stretchy":703},[701,3857,853],{},[691,3859,3860,3862],{},[714,3861,3744],{},[714,3863,3349],{},[701,3865,853],{},[714,3867,3868],{},"16",[701,3870,1309],{},[714,3872,3873],{},"6",[788,3875],{},[788,3877],{"width":1322},[701,3879,704],{"stretchy":703},[589,3881,3882,3884,3886],{},[592,3883,594],{"mathvariant":1329},[592,3885,1332],{"mathvariant":1329},[592,3887,1335],{"mathvariant":1329},[788,3889],{"width":1338},[714,3891,3310],{},[701,3893,709],{"stretchy":703},[596,3895,3896],{"encoding":598},"2^{\\varphi(10)} = 2^4 = 16 \\equiv 6 \\pmod{10}",[573,3898,3900,3956,4000,4018,4033],{"className":3899,"ariaHidden":603},[602],[573,3901,3903,3906,3947,3950,3953],{"className":3902},[607],[573,3904],{"className":3905,"style":752},[611],[573,3907,3909,3912],{"className":3908},[616],[573,3910,3744],{"className":3911},[616],[573,3913,3915],{"className":3914},[739],[573,3916,3918],{"className":3917},[743],[573,3919,3921],{"className":3920},[747],[573,3922,3924],{"className":3923,"style":752},[751],[573,3925,3926,3929],{"style":755},[573,3927],{"className":3928,"style":760},[759],[573,3930,3932],{"className":3931},[764,765,766,767],[573,3933,3935,3938,3941,3944],{"className":3934},[616,767],[573,3936,699],{"className":3937},[616,617,767],[573,3939,704],{"className":3940},[777,767],[573,3942,3310],{"className":3943},[616,767],[573,3945,709],{"className":3946},[784,767],[573,3948],{"className":3949,"style":874},[788],[573,3951,853],{"className":3952},[878],[573,3954],{"className":3955,"style":874},[788],[573,3957,3959,3962,3991,3994,3997],{"className":3958},[607],[573,3960],{"className":3961,"style":2561},[611],[573,3963,3965,3968],{"className":3964},[616],[573,3966,3744],{"className":3967},[616],[573,3969,3971],{"className":3970},[739],[573,3972,3974],{"className":3973},[743],[573,3975,3977],{"className":3976},[747],[573,3978,3980],{"className":3979,"style":2561},[751],[573,3981,3982,3985],{"style":755},[573,3983],{"className":3984,"style":760},[759],[573,3986,3988],{"className":3987},[764,765,766,767],[573,3989,3349],{"className":3990},[616,767],[573,3992],{"className":3993,"style":874},[788],[573,3995,853],{"className":3996},[878],[573,3998],{"className":3999,"style":874},[788],[573,4001,4003,4006,4009,4012,4015],{"className":4002},[607],[573,4004],{"className":4005,"style":803},[611],[573,4007,3868],{"className":4008},[616],[573,4010],{"className":4011,"style":874},[788],[573,4013,1309],{"className":4014},[878],[573,4016],{"className":4017,"style":874},[788],[573,4019,4021,4024,4027,4030],{"className":4020},[607],[573,4022],{"className":4023,"style":803},[611],[573,4025,3873],{"className":4026},[616],[573,4028],{"className":4029},[788,1529],[573,4031],{"className":4032,"style":1533},[788],[573,4034,4036,4039,4042,4051,4054,4057],{"className":4035},[607],[573,4037],{"className":4038,"style":933},[611],[573,4040,704],{"className":4041},[777],[573,4043,4045],{"className":4044},[616],[573,4046,4048],{"className":4047},[616],[573,4049,1553],{"className":4050},[616,1552],[573,4052],{"className":4053,"style":1557},[788],[573,4055,3310],{"className":4056},[616],[573,4058,709],{"className":4059},[784],", nicht ",[573,4062,4064,4077],{"className":4063},[576],[573,4065,4067],{"className":4066},[580],[582,4068,4069],{"xmlns":584},[586,4070,4071,4075],{},[589,4072,4073],{},[714,4074,716],{},[596,4076,716],{"encoding":598},[573,4078,4080],{"className":4079,"ariaHidden":603},[602],[573,4081,4083,4086],{"className":4082},[607],[573,4084],{"className":4085,"style":803},[611],[573,4087,716],{"className":4088},[616],[1601,4090,4091,4183,4184,4235,4236,4060,4264,1919],{},[569,4092,4093,4182],{},[573,4094,4096,4124],{"className":4095},[576],[573,4097,4099],{"className":4098},[580],[582,4100,4101],{"xmlns":584},[586,4102,4103,4121],{},[589,4104,4105,4107,4109,4111,4113,4115,4117,4119],{},[592,4106,699],{},[701,4108,704],{"stretchy":703},[592,4110,594],{},[701,4112,709],{"stretchy":703},[701,4114,853],{},[592,4116,594],{},[701,4118,712],{},[714,4120,716],{},[596,4122,4123],{"encoding":598},"\\varphi(m) = m - 1",[573,4125,4127,4154,4173],{"className":4126,"ariaHidden":603},[602],[573,4128,4130,4133,4136,4139,4142,4145,4148,4151],{"className":4129},[607],[573,4131],{"className":4132,"style":933},[611],[573,4134,699],{"className":4135},[616,617],[573,4137,704],{"className":4138},[777],[573,4140,594],{"className":4141},[616,617],[573,4143,709],{"className":4144},[784],[573,4146],{"className":4147,"style":874},[788],[573,4149,853],{"className":4150},[878],[573,4152],{"className":4153,"style":874},[788],[573,4155,4157,4161,4164,4167,4170],{"className":4156},[607],[573,4158],{"className":4159,"style":4160},[611],"height:0.6667em;vertical-align:-0.0833em;",[573,4162,594],{"className":4163},[616,617],[573,4165],{"className":4166,"style":789},[788],[573,4168,712],{"className":4169},[793],[573,4171],{"className":4172,"style":789},[788],[573,4174,4176,4179],{"className":4175},[607],[573,4177],{"className":4178,"style":803},[611],[573,4180,716],{"className":4181},[616]," einsetzen."," Das stimmt nur für Primzahlen. Für ",[573,4185,4187,4205],{"className":4186},[576],[573,4188,4190],{"className":4189},[580],[582,4191,4192],{"xmlns":584},[586,4193,4194,4202],{},[589,4195,4196,4198,4200],{},[592,4197,594],{},[701,4199,853],{},[714,4201,2610],{},[596,4203,4204],{"encoding":598},"m = 100",[573,4206,4208,4226],{"className":4207,"ariaHidden":603},[602],[573,4209,4211,4214,4217,4220,4223],{"className":4210},[607],[573,4212],{"className":4213,"style":612},[611],[573,4215,594],{"className":4216},[616,617],[573,4218],{"className":4219,"style":874},[788],[573,4221,853],{"className":4222},[878],[573,4224],{"className":4225,"style":874},[788],[573,4227,4229,4232],{"className":4228},[607],[573,4230],{"className":4231,"style":803},[611],[573,4233,2610],{"className":4234},[616]," ist der Exponent ",[573,4237,4239,4252],{"className":4238},[576],[573,4240,4242],{"className":4241},[580],[582,4243,4244],{"xmlns":584},[586,4245,4246,4250],{},[589,4247,4248],{},[714,4249,2650],{},[596,4251,2650],{"encoding":598},[573,4253,4255],{"className":4254,"ariaHidden":603},[602],[573,4256,4258,4261],{"className":4257},[607],[573,4259],{"className":4260,"style":803},[611],[573,4262,2650],{"className":4263},[616],[573,4265,4267,4281],{"className":4266},[576],[573,4268,4270],{"className":4269},[580],[582,4271,4272],{"xmlns":584},[586,4273,4274,4279],{},[589,4275,4276],{},[714,4277,4278],{},"99",[596,4280,4278],{"encoding":598},[573,4282,4284],{"className":4283,"ariaHidden":603},[602],[573,4285,4287,4290],{"className":4286},[607],[573,4288],{"className":4289,"style":803},[611],[573,4291,4278],{"className":4292},[616],[1601,4294,4295,4341,4342,4385,4386,4520,4521,1919],{},[569,4296,4297,4340],{},[573,4298,4300,4319],{"className":4299},[576],[573,4301,4303],{"className":4302},[580],[582,4304,4305],{"xmlns":584},[586,4306,4307,4317],{},[589,4308,4309,4311,4313,4315],{},[592,4310,699],{},[701,4312,704],{"stretchy":703},[592,4314,594],{},[701,4316,709],{"stretchy":703},[596,4318,1009],{"encoding":598},[573,4320,4322],{"className":4321,"ariaHidden":603},[602],[573,4323,4325,4328,4331,4334,4337],{"className":4324},[607],[573,4326],{"className":4327,"style":933},[611],[573,4329,699],{"className":4330},[616,617],[573,4332,704],{"className":4333},[777],[573,4335,594],{"className":4336},[616,617],[573,4338,709],{"className":4339},[784]," für die kleinste Periode halten."," Der Satz garantiert nur, dass ",[573,4343,4345,4364],{"className":4344},[576],[573,4346,4348],{"className":4347},[580],[582,4349,4350],{"xmlns":584},[586,4351,4352,4362],{},[589,4353,4354,4356,4358,4360],{},[592,4355,699],{},[701,4357,704],{"stretchy":703},[592,4359,594],{},[701,4361,709],{"stretchy":703},[596,4363,1009],{"encoding":598},[573,4365,4367],{"className":4366,"ariaHidden":603},[602],[573,4368,4370,4373,4376,4379,4382],{"className":4369},[607],[573,4371],{"className":4372,"style":933},[611],[573,4374,699],{"className":4375},[616,617],[573,4377,704],{"className":4378},[777],[573,4380,594],{"className":4381},[616,617],[573,4383,709],{"className":4384},[784]," eine Periode ist. Oft reicht ein kleinerer Exponent: ",[573,4387,4389,4431],{"className":4388},[576],[573,4390,4392],{"className":4391},[580],[582,4393,4394],{"xmlns":584},[586,4395,4396,4428],{},[589,4397,4398,4404,4406,4408,4410,4412,4414,4422,4424,4426],{},[691,4399,4400,4402],{},[714,4401,3425],{},[714,4403,3349],{},[701,4405,1309],{},[714,4407,716],{},[788,4409],{},[788,4411],{"width":1322},[701,4413,704],{"stretchy":703},[589,4415,4416,4418,4420],{},[592,4417,594],{"mathvariant":1329},[592,4419,1332],{"mathvariant":1329},[592,4421,1335],{"mathvariant":1329},[788,4423],{"width":1338},[714,4425,2610],{},[701,4427,709],{"stretchy":703},[596,4429,4430],{"encoding":598},"7^4 \\equiv 1 \\pmod{100}",[573,4432,4434,4478,4493],{"className":4433,"ariaHidden":603},[602],[573,4435,4437,4440,4469,4472,4475],{"className":4436},[607],[573,4438],{"className":4439,"style":2561},[611],[573,4441,4443,4446],{"className":4442},[616],[573,4444,3425],{"className":4445},[616],[573,4447,4449],{"className":4448},[739],[573,4450,4452],{"className":4451},[743],[573,4453,4455],{"className":4454},[747],[573,4456,4458],{"className":4457,"style":2561},[751],[573,4459,4460,4463],{"style":755},[573,4461],{"className":4462,"style":760},[759],[573,4464,4466],{"className":4465},[764,765,766,767],[573,4467,3349],{"className":4468},[616,767],[573,4470],{"className":4471,"style":874},[788],[573,4473,1309],{"className":4474},[878],[573,4476],{"className":4477,"style":874},[788],[573,4479,4481,4484,4487,4490],{"className":4480},[607],[573,4482],{"className":4483,"style":803},[611],[573,4485,716],{"className":4486},[616],[573,4488],{"className":4489},[788,1529],[573,4491],{"className":4492,"style":1533},[788],[573,4494,4496,4499,4502,4511,4514,4517],{"className":4495},[607],[573,4497],{"className":4498,"style":933},[611],[573,4500,704],{"className":4501},[777],[573,4503,4505],{"className":4504},[616],[573,4506,4508],{"className":4507},[616],[573,4509,1553],{"className":4510},[616,1552],[573,4512],{"className":4513,"style":1557},[788],[573,4515,2610],{"className":4516},[616],[573,4518,709],{"className":4519},[784],", obwohl ",[573,4522,4524,4547],{"className":4523},[576],[573,4525,4527],{"className":4526},[580],[582,4528,4529],{"xmlns":584},[586,4530,4531,4545],{},[589,4532,4533,4535,4537,4539,4541,4543],{},[592,4534,699],{},[701,4536,704],{"stretchy":703},[714,4538,2610],{},[701,4540,709],{"stretchy":703},[701,4542,853],{},[714,4544,2650],{},[596,4546,2653],{"encoding":598},[573,4548,4550,4577],{"className":4549,"ariaHidden":603},[602],[573,4551,4553,4556,4559,4562,4565,4568,4571,4574],{"className":4552},[607],[573,4554],{"className":4555,"style":933},[611],[573,4557,699],{"className":4558},[616,617],[573,4560,704],{"className":4561},[777],[573,4563,2610],{"className":4564},[616],[573,4566,709],{"className":4567},[784],[573,4569],{"className":4570,"style":874},[788],[573,4572,853],{"className":4573},[878],[573,4575],{"className":4576,"style":874},[788],[573,4578,4580,4583],{"className":4579},[607],[573,4581],{"className":4582,"style":803},[611],[573,4584,2650],{"className":4585},[616],{"title":4587,"searchDepth":4588,"depth":4588,"links":4589},"",2,[4590,4591,4592],{"id":562,"depth":4588,"text":563},{"id":2522,"depth":4588,"text":2523},{"id":3717,"depth":4588,"text":3718},[4594,4598,4601,4603,4605,4607,4609,4611,4613,4615,4617,4619],{"lhs":4595,"rhs":716,"where":4596},"mod(a^4, 10)",{"a":4597},3,{"lhs":4595,"rhs":716,"where":4599},{"a":4600},7,{"lhs":4602,"rhs":2650},"100*(1 - 1\u002F2)*(1 - 1\u002F5)",{"lhs":4604,"rhs":2548},"3*40 + 3",{"lhs":4606,"rhs":716},"mod(bignumber(3)^40, 100)",{"lhs":4608,"rhs":2933},"mod(bignumber(3)^123, 100)",{"lhs":4610,"rhs":2933},"3^3",{"lhs":4612,"rhs":3349},"10*(1 - 1\u002F2)*(1 - 1\u002F5)",{"lhs":4614,"rhs":3425},"mod(3^3, 10)",{"lhs":4616,"rhs":716},"mod(3*7, 10)",{"lhs":4618,"rhs":3873},"mod(2^4, 10)",{"lhs":4620,"rhs":716},"mod(7^4, 100)","m \\in \\mathbb{N},\\ m \\geq 2,\\ a \\in \\mathbb{Z},\\ \\operatorname{ggT}(a, m) = 1","\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>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\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:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\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>a\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">Z\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a \\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:0.5782em;vertical-align:-0.0391em;\">\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:0.8833em;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 mathvariant=\"normal\">ggT\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{ggT}(a, m) = 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:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">ggT\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\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\">m\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","euler-theorem","a^{\\varphi(m)} \\equiv 1 \\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>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo>≡\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\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 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