Question 59
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:
$$
\operatorname{gcd}(A, B, C, D)=1
$$
Here $\operatorname{gcd}(A, B, C, D)$ is the largest positive integer that divides all of $A, B, C$ and $D$.
$$
\operatorname{gcd}(A, B, C, D)=1
$$
Here $\operatorname{gcd}(A, B, C, D)$ is the largest positive integer that divides all of $A, B, C$ and $D$.