Assume a is not divisible by 10. (otherwise the problem is trivial). <span>Define R(m) to be the remainder of a^m when divided by 10. </span> <span>R can take on one of 9 possible values, namely, 1,2,...,9. </span> <span>Now, consider R(1),R(2),......R(10). At least 2 of them must have the sames value (by the Pigeonhole Principle), say R(i) = R(j) ( j>i ) </span> <span>Then, a^j - a^i is divisible by 10.</span>