Let a, b, c, d be integers, and let H be the subgroup of Z × Z generated by (a, b) and (c, d). Thus H is the set of all elements of the form (ma, mb) + (nc, nd) where m, n ∈ Z.

(a) Suppose b = 0, c = 0. Prove that (Z × Z)/H is isomorphic to Z/aZ × Z/dZ.

(b) Suppose (a, b) = (10, 12) and (c, d) = (4, 4). Find a product of cyclic groups isomorphic to (Z × Z)/H .

(c) Determine, in terms of a, b, c, d, when (Z × Z)/H is ﬁnite. When it is ﬁnite, ﬁnd its order.

I can see intuitively that the statements for a and b are right but I can't come up with any method of proof

could anyone help me?? plelase ?