Question 20
Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_{i} \cdot F_{j}$, where $\left(F_{n}\right)_{n=1}^{\infty}$ is the Fibonacci sequence defined by $F_{1}=F_{2}=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for all $n \geq 3$.