I have no idea with this question. Please see more detail in the attachment, thank you.

7. (a) Let n, m E N . Prove that if god(m, n) = 1 then Zim X Zn ~ Zmn. (b) Give an example of positive integers n, m such that Zn X Zim F Zinm, and prove that these two groups are not isomorphic. (c) Prove the converse to part (a), i.e., if Zim X Zn ~ Zmn, then god(n, m) = 1. (Hint : Assume god(n, m) = d * 1. Show that Zm X Zn is not cyclic. )