M is a positive integer ≥ 2 such that 3^M + 4^M is divisible by M.

Is M always divisible by 7?

If so, prove it. Otherwise, provide a counterexample.

1. 3^M+4^M is odd, so even M merit no further consideration.

2. By reason of addition/multiplication of powers, we need further only check odd prime M.

3. If M is odd prime, then we have 3^(M-1) congruent to 1 and 4^(M-1) congruent to 1, modM (Fermat).

4. Hence, 3^M is worth 3, mod5 or larger and 4^M is worth 4, mod5 or larger.

5. Accordingly, M is always divisible by 7.

Edited on November 13, 2010, 12:40 pm

Posted by broll on 2010-11-13 12:39:18