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Seven Divisibility Settlement (Posted on 2010-11-13) Difficulty: 3 of 5
M is a positive integer ≥ 2 such that 3M + 4M is divisible by M.

Is M always divisible by 7?

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

No Solution Yet Submitted by K Sengupta    
Rating: 2.0000 (2 votes)

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Possible Solution | Comment 1 of 3

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

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