Question 20
Consider the product $P_n = 1! \cdot 2! \cdot 3! \cdot \dots \cdot n!$, where $n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n$, for every positive integer $n$.
a) Find all possible values of positive integers $m$ such that $\frac{P_{2020}}{m!}$ is a perfect square.
b) Prove that there exist infinitely many values of $n$ such that $\frac{P_n}{m!}$ is a perfect square for at least two positive integers $m$.
a) Find all possible values of positive integers $m$ such that $\frac{P_{2020}}{m!}$ is a perfect square.
b) Prove that there exist infinitely many values of $n$ such that $\frac{P_n}{m!}$ is a perfect square for at least two positive integers $m$.