Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
Problem
Source: 2021 Czech-Polish-Slovak Match Junior, team p3 CPSJ
Tags: number theory, greatest common divisor, least common multiple, LCM, GCD