An algebra problem by Aly Ahmed

We assume that $x$ is a positive integer (since it is not given).

For all positive integers $x$ , the unit's place digit of $3^x$ can be $1,3,7,9$ , of $4^x$ can be $4,6$ , of $5^x$ can be $5$ , and of $6^x$ can be $6$ . So, the unit's place digit of $3^x+4^x+5^x$ has to be $6$ (since $3,5$ are odd and $4$ is even, the unit's place digit of the L. H. S. can be even only). So, the only possibility is that $x=\boxed 3$ (by direct check).