#1**0 **

Write in \(34X1_5\) in base 10, which would be \((3*5^3+4*5^2+X*5^1+1*5^0)_{10}\)

Simplifying that gives \(375+100+5X+1=476+5X\)

Now, the next highest integer after 476 that is divisible by 31 is 496 (You can find this by doing 476/31 and then if it is not a whole number adding 1 to find the next number). Since we are looking to set this equal to 496, because it's the next digit divisible by 31, we have \(476+5X=496 5X=496-476 5X=20 X=\boxed{4}\)

Also, not that if you found X>4 it would be an extraneous or just wrong solution, because it asks for the "digit" X, so it would have to be one of 0, 1, 2, 3, 4, because it is in base 5.

RiemannIntegralzzz Apr 2, 2021