Problem - 4185
Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.
Taking MOD $4$ on both sides gives $(-1)^x + 0 \equiv 1^z\pmod{4}$. It can only hold if $x$ is even.
Next, taking MOD $3$ on both sides yields $0 + 1^y \equiv (-1)^z\pmod{3}$. Hence, $z$ must be even.