Let $p_1,p_2,q_1,q_2$ be irreducible elements in integral domain $R$ such that none are associates to any of the others and $p_1p_2=q_1q_2$.
Prove that $p_1q_1q_2$ and $p_1p_2q_1$ do not have a greatest common divisor.
I have no idea to approach this problem.
It will be started with having gcd and show contradiction. But there are not much theorem to use show the question with integral domain. Help me!!