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Powers of Ten (Posted on 2015-07-14) Difficulty: 3 of 5
Does the arithmetic sequence 1, 2188, 4375, 6562, ...... contain infinitely many powers of 10?
Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Pwerful answer (spoiler) | Comment 1 of 7
Well, of course it does.

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 powers 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, using excel, I see that 10^244 = 1 mod 2187, so the cycle has length 244.  10^n = 1 mod 2187  for n = 244k, where k is any integer.

Edited on July 14, 2015, 2:57 pm
  Posted by Steve Herman on 2015-07-14 11:58:09

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