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Well, obviously it does contain infinitely many powers of 10.
We are looking for values 10^N which equal 1 mod 2187.
Because 10^0 = 1, mod N, the sequence 10^N must eventually equal 1 for some higher N, because power
cycle with respect to any mod. There is nothing magic about 2187 or 10 for that matter. For any k and s, the arithmetic series 1, 1+k, 1+2k, ... contains
infinitely many powers of s for any positive integers
k and s.
In this case, we observe that 10^243 = 1 mod 2187, so the cycle has length 243. 10^n = 1 mod 2187 for n = 243k, where k is any integer. |
