EGMO 2024 Problem 3

We call a positive integer $n$ peculiar if, for any positive divisor $d$ of $n$, the integer $d(d + 1)$ divides $n(n + 1)$. Prove that for any four different peculiar positive integers $A, B, C$ and $D$, the following holds:

$$\gcd(A, B, C, D) = 1.$$

Here $\gcd(A, B, C, D)$ is the largest positive integer that divides all of $A, B, C$ and $D$.