I came across a problem in Niven's number theory text (problem 51 on page 20) that asks the following:
Show that if $(a, b) = 1$ and $p$ is an odd prime, then $\left(a + b, \frac{a^p + b^p}{a + b}\right) = 1 \text{ or } p.$
I am not asking for a solution to this problem; instead, I'm trying to understand why $a^p + b^p$ would always be divisible by $a + b$ given the above conditions. Does anyone have any insights as to why this would be true? Where (if at all) do we use the conditions that $(a, b) = 1$ and $p$ is an odd prime?