Find all generators for $\Bbb Z_4\times \Bbb Z_5$.

Question

I'm looking for all generators $\Bbb Z_4\times \Bbb Z_5$.

I know I could do this by writing out all the elements and then going through and finding the generators by trial and error, but is there a faster way?

Answer 1

Since $4$ is composite, the generators of $\Bbb Z_4$ are $[1]_4$ and $[3]_4$ as both $1$ and $3$ are coprime to $4$; however, $5$ is prime, so $[1]_5, [2]_5, [3]_5, $ and $[4]_5$ are generators of $\Bbb Z_5$.

But $4$ and $5$ are coprime. What can you conclude from this?

The number of generators of $\Bbb Z_4\times \Bbb Z_5$ is $2\times 4=8$.

Can you continue from here?