Let and be nontrivial abelian groups. Prove that is divisible if and only if and are divisible.
Suppose is divisible, and let . Since is divisible, for every there exists an element such that . Thus for every there exists such that ; hence is divisible. By a similar argument, is divisible.
Suppose and are divisible, and let . Let . Since and are divisible, there exist and such that and ; then . Thus is divisible.