Cyclic congruencies Suppose $a$ and $b$ are positive integers. Set $x_n := a^n $ modulo $b$. Consider $\{x_n\}$. My question is: is it always true that this sequence must be cyclic?  
I am guessing there is some algebraic reason why this is true, but I don't know enough algebra to think of it myself.

The answer is no.  Take $a=2, b=4$.  The sequence $a, a^2, a^3, \ldots$ (mod $4$) is $$2, 0, 0,\ldots$$
However, the sequence does have to be eventually cyclic.
