jmc

The base 5 number $34x{1}_5$ is equal to $3\cdot 5^3+4\cdot 5^2+x\cdot 5+1=5x+476$. The base-5 digit $x$ must be 0, 1, 2, 3, or 4. Among these values, only $x=\boxed{4}$ makes $5x+476$ divisible by 31.