KöMaL 2019 C1524

Let $N$ and $M$ be positive integers, and let $p$ and $q$ denote different primes. Assume that $N+M$ is a five-digit number, $N$ is divisible by $p$ and the number of its divisors is $q$, while $M$ is divisible by $q$ and the number of its divisors is $p$. Determine all possible values of $N$ and $M$.