Is it true that $a^{360m + 1} + b^{360n + 1} \equiv 0 (\mod 475) \Leftrightarrow a + b \equiv 0 (\mod 475)$?
It is obvious that we only need to check if $a + b \equiv 0 (\mod 475)$ implied $a^{360m + 1} + b^{360n + 1} \equiv 0 (\mod 475)$ or not.
With the notice that $360 = \varphi (475)$, we are done with the case $(a,475) = 1$ (which leads to $(b,475) = 1$).
But when dealing with the case $(a,475) = 5$, I have some doubt. In this case, $a = 5k$, $b = 5l$ where $(k,5) = 1$, $(l,5) = 1$. A counter-example where $a + b \equiv 0 (\mod 25)$ but $a^{360m + 1} + b^{360n + 1} \not\equiv 0 (\mod 25)$ could have existed.
I'm not sure how to proceed as I don't even know if the statement itself was true at the beginning. Please give me some hint. Thank you.