Positive integers satisfying $a + b = c + d$ and $ab\vert cd$

Question

I need to determine wheather there exist odd integers $a,b,c,d>1$ satisfying $a + b = c + d$ and $ab\mid cd$.

(Excluding the trivial cases, $a=c$ or $a=d$)

Any help is appreciated.

I have tried elementary methods but none works. I am searching for a more complicated method to help me determine but can't seem to find.

Thanks!

Answer 1

Take $(a,b,c,d)=(3,18k+9,6k+3,12k+9)$ for $k\in \mathbb N$ and $3|k$, to get infinitely many solutions.