Suppose that $\displaystyle x,y,z$ are integers, none of which is divisible by $\displaystyle 5$.

Show that $\displaystyle x^5 + y^5 \neq z^5$. Hint: work modulo $\displaystyle 25$.

I can only think of these congruences by Fermat's Little Theorem.

$\displaystyle x^5 \equiv x (\mod 5)$

$\displaystyle y^5 \equiv y (\mod 5)$

$\displaystyle z^5 \equiv z (\mod 5)$.

How can I get congruences in modulo 25 and prove the statement?

Is it proved by assuming that $\displaystyle x^5 + y^5 = z^5$ and then show a contradiction like the proof for $\displaystyle n = 5$ in

https://en.wikipedia.org/wiki/Proof_of_Fermat's_Last_Theorem_for_specific_expone nts ?