(a) Let \(m, n\) be coprime integers, and suppose \(a\) is an integer which is divisible by both \(m\) and \(n\). Prove that \(m n\) divides \(a\).

(b) Show that the conclusion of part (a) is false if \(m\) and \(n\) are not coprime (i.e., show that if \(m\) and \(n\) are not coprime, there exists an integer \(a\) such that \(m \mid a\) and \(n \mid a\), but \(m n\) does not divide \(a\) ).

(c) Show that if \(\operatorname{hcf}(x, m)=1\) and \(\operatorname{hcf}(y, m)=1\), then \(\operatorname{hcf}(x y, m)=1\).