Prove that if a is a natural number, then there exist two unequal natural numbers k and l for which ak-al is divisible by 10.

Assume a is not divisible by 10. (otherwise the problem is trivial).

Define R (m) to be the remainder of a^m when divided by 10.

R can take on one of 9 possible values, namely, 1,2, ...,9.

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)

Then, a^j - a^i is divisible by 10.

Divisible by 10 ...