The statement of the problem is equivalent to saying that an integer $n \ge 5$ is composite if and only if there exists a representation of $n$ as the sum of four positive integers such that $ab = cd$.

($\Leftrightarrow$) If $ab = cd$ and $n = a+b+c+d$, then
$$an = a^2 + ab + ac + ad = a^2 + cd + ac + ad = (a+c)(a+d), (*)$$
hence $n \mid (a+c)(a+d)$.

($\Rightarrow$) If $n = pq$, with $p, q \ge 2$, is a composite number, we can choose $a = 1$, $b = (p-1)(q-1)$, $c = p-1$, $d = q-1$. This choice can easily be guessed by writing $(*)$ with $a = 1$.