If $p~ \mid~ m^p + n^p$, prove $p~ \mid~ \frac{m^p + n^p}{m+n}$.

Question

If $p \mid m^p + n^p$, and $p$ is a prime greater than $2$, prove $$p \mid \frac{m^p + n^p}{m+n}.$$

No clue how to start. Clearly $p \mid m + n$, but then what. I feel very less information is given. Yet primes are full of surprises.

Thanks in advance.

Answer 1

Hint : see $m \equiv -n$ $(\text {mod} p)$. Now factorise the part $\dfrac{m^p + n^p}{m + n}$ and then apply $m \equiv -n$ $(\text {mod} p)$.

Answer 2

You can add arbitrary terms of the form $pm^an^b$ to the right side, and it will remain true. Then try using binomial theorem.