2021 WSMO Team Round Problems/Problem 12

Problem

Choose three integers $x,y,z$ randomly and independently from the nonnegative integers. The probability that the sum of the factors of $2^x3^y5^z$ is divisible by $6$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Proposed by pinkpig

Solution