The positive powers of 2 mod 5^m cycle with period 4*5^(m-1), which you can prove by showing that 2 is a primitive root mod powers of 5. I want to prove that the positive powers of two, mod 10^m, also cycle with this same period. How do I go from this powers of 5 result to powers of 10?(adsbygoogle = window.adsbygoogle || []).push({});

The Chinese Remainder Theorem (CRT) obviously came to mind, so I considered the powers of 2 mod 2^m. Starting at 2^m, they cycle with period 1 (they are always 0). The answer, I'm sure you're going to say, is to multiply the two periods: 1 x 4 = 4. But what specifically about the CRT lets me do that? The statements of the CRT I've seen talk about the residues, not the periods.

Maybe I need to invoke an underlying theorem from group theory instead?

Thanks.

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# Cycle Length of the Positive Powers of Two Mod Powers of Ten

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