External direct products: abstract algebra?

Question

How do you prove that Z_36 x Z_30 is congruent to Z_60 x Z_18??

Answer 1

p and q are relatively prime if and only if Z_{pq} is isomorphic to Z_p x Z_q. if you need proof of this fact, see theorem 1.1 in http://www.maths.bris.ac.uk/~rp3959/dirprodcycgrps.pdf so Z_36 = Z_4 x Z_9 and Z_30 = Z_2 x Z_3 x Z_5. we also have Z_60 = Z_4 x Z_3 x Z_5 and Z_18 = Z_2 x Z_9. so both direct products are isomorphic to
Z_2 x Z_3 x Z_4 x Z_5 x Z_9.

Answer 2

Define a mapping from one to the other and show that your mapping is an isomorphism.