$(\mathbb Z_4\times\mathbb Z_6)/\langle(2,3)\rangle$ is isomorphic to $\mathbb Z_2\times\mathbb Z_6$ or $\mathbb Z_{12}$? [duplicate]

I found that $(\mathbb Z_4\times\mathbb Z_6)/\langle(2,3)\rangle$ should be isomorphic to $\mathbb Z_4\times\mathbb Z_3$.

But I cannot construct a homomorphism $\phi:\mathbb Z_4\times\mathbb Z_6\to\mathbb Z_4\times\mathbb Z_3$ such that $\ker\phi=\langle(2,3)\rangle$.

A function such that $(a,b)\to(a,b)$ when $b\leq2$ and $(a,b)\to(a+2,b)$ otherwise seems like the function I want, but it's formula is too complicated.

Can anyone help me construct $\phi$ in efficient manner?

Thanks.

Solution 1:

Your formula looks right to me except the second number in the target needs to be in the range $0$ to $2$. One way to express it is $(a,b) \to (a+2(b/3), b \bmod 3)$ where the divide is integer divide.