I need some help finding a mapping between $\mathbb{Z}_3\times\mathbb{Z}_5$ and $\mathbb{Z}_{15}$. I'm a bit confused as to how multiplication in different modulo works, and how I should "connect" the elements.
What I have so far is: for $\mathbb{Z}_3\times\mathbb{Z}_5$, we have {(1x1), (1x2), (1x3), (1x4), (1x5), (2x1), (2x2), (2x3), (2x4), (2x5), (3x1), (3x2), (3x3), (3x4), (3x5)}, and for $\mathbb{Z}_{15}$ we have $\{1,\; 2,\; \ldots ,\; 15\}$. Since they both have the same number of elements, a bijection is possible. However, I just can't seem to come up with an isomorphic mapping. Is it just the $n^{th}$ element mapped to the $n^{th}$ element between the two sets? Thank you for your help.
Hint 1: A homomorphism between cyclic groups is defined by where it sends the generator. From there, you need to prove injectivity and surjectivity. That the two groups in the product have relatively prime order is crucial.