Can someone just explain to me the basic process of what is going on here? I understand everything until we start adding 1's then after that it all goes to hell. I just need some guidance. The Problem with the solution is attached.
Thanks in advance..
Answer
Note that 645=3\cdot 5\cdot 43. Then
11^{644}\equiv (-1)^{2\cdot 322}\equiv 1^{322}\equiv 1\pmod 3 11^{644}\equiv 1^{644}\equiv 1\pmod 5 For the last modulus, we should determine the order of 11\pmod{43}. To this end we first try 11^q for q\mid p-1: 11^2\equiv 35\equiv -8, 11^3\equiv -8\cdot 11\equiv -2, 11^7\equiv (-8)^2\cdot(-2)\equiv -128\equiv 1. So with this 11^{644}\equiv 11^{7\cdot 46}\equiv 1^{46}\equiv 1\pmod{43} and so by the Chinese Remainder Theorem also 11^{644}\equiv 1\pmod {645}
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