elements of order 15 in $Z_5 \times Z_{15}$ I know that for an element $(a,b) \in \mathbb{Z}_j \times \mathbb{Z}_k$, the element will have order $lcm(m,n)$ where m is the order of a in $\mathbb{Z}_j$ and n is the order of b in $\mathbb{Z}_k$. So if I am supposed to instead find such a,b do I just go through all possibilities for m and n that give 15 then check if a and b can have such an order in their respective quotient groups? 

I am ultimately trying to find the number of elements of order 15 in $\mathbb{Z}_5 \times \mathbb{Z}_{15}$
$$\mathbb{Z}_5\times \mathbb{Z}_{15}\cong \mathbb{Z}_5\times \mathbb{Z}_5\times \mathbb{Z}_3$$

Hence, we need to select an element of order $5$ from $\mathbb{Z}_5\times \mathbb{Z}_5$, and combine with an element of order $3$ from $\mathbb{Z}_3$.  There are $24$ ways to do the former (anything but the identity), and $2$ ways to do the latter.  Hence the answer you seek is $48$.
$Z_5 \times Z_{15}=&lt;f1,f2,f3&gt;$
$f2, f1f2, f2^2, f2f3, f1^2f2, f1f2^2, f1f2f3, f2f3^2, f1^3f2, f1^2f2^2, f1^2f2f3, f1f2^2f3,f1f2f3^2, f2^2f3^2, f1^4f2, f1^3f2^2, f1^3f2f3, f1^2f2^2f3, f1^2f2f3^2, f1f2^2f3^2,f1f2f3^3, f2^2f3^3, f2f3^4, f1^4f2^2, f1^4f2f3, f1^3f2^2f3, f1^3f2f3^2, f1^2f2^2f3^2, f1^2f2f3^3, f1f2^2f3^3, f1f2f3^4, f2^2f3^4, f1^4f2^2f3, f1^4f2f3^2, f1^3f2^2f3^2,f1^3f2f3^3, f1^2f2^2f3^3, f1^2f2f3^4, f1f2^2f3^4, f1^4f2^2f3^2, f1^4f2f3^3, f1^3f2^2f3^3,f1^3f2f3^4, f1^2f2^2f3^4, f1^4f2^2f3^3, f1^4f2f3^4, f1^3f2^2f3^4, f1^4f2^2f3^4 $ are of order 15