Suppose there are four integers $p,q,r,s$ such that
$$p \cdot q = r \cdot s$$
and $p,q,r,s$ are all Carmichael numbers. Obviously there are trivial solutions, namely $p=r,q=s$ and $p=s,q=r$. My question is: can you find any non-trivial solution? Can you prove that only trivial solutions exist?
I have found four different Carmichael numbers \begin{align*} a & = 7\cdot 31\cdot 73, \\ b & = 7\cdot 13\cdot 31\cdot 61, \\ c & = 37\cdot 73\cdot 181, \\ d & = 13\cdot 37\cdot 61\cdot 181, \\ \end{align*} which satisfy $ad=bc=7\cdot 13\cdot 31\cdot 37\cdot 61\cdot 73\cdot 181$.