Isomorphism between the groups $\mathbb Z/(a) \times \mathbb Z / (b)$ and $\mathbb Z/(m) \times \mathbb Z / (n) $ Let $a,b$ be non-zero integers with GCD $m$ and LCM $n$. How to show that $\mathbb Z/(a) \times \mathbb Z / (b) \cong \mathbb Z/(m) \times \mathbb Z / (n) $  (isomorphic as groups i.e. $\mathbb Z$-modules ) ? 
I am not sure what the canonical map is ... Please help 
 A: Here's a low-tech approach. Start with a simple case: suppose $a$ and $b$ are powers of some prime $p$, $a = p^i$ and $b = p^j$. Then $m = p^{\min(i, j)}$ and $n = p^{\max(i, j)}$. So $m$ and $n$ are the same as $a$ and $b$ (perhaps swapped), and there's clearly an isomorphism as required.
Now suppose there are two different primes involved, $p_0$ and $p_1$: $a = {p_0}^{i_0} {p_1}^{i_1}$, $b = {p_0}^{j_0} {p_1}^{j_1}$. Intuitively, because $p_0$ and $p_1$ are relatively prime, they operate independently. So $\Bbb{Z}/(a) \simeq \Bbb{Z}/({p_0}^{i_0}) \times \Bbb{Z}/({p_1}^{i_1})$ and $\Bbb{Z}/(b) \simeq \Bbb{Z}/({p_0}^{j_0}) \times \Bbb{Z}/({p_1}^{j_1})$.  Also, $m = {p_0}^{\min(i_0, j_0)} {p_1}^{\min(i_1, j_1)}$ and $n = {p_0}^{\max(i_0, j_0)} {p_1}^{\max(i_1, j_1)}$, so $\Bbb{Z}/(m) \simeq \Bbb{Z}/({p_0}^{\min(i_0, j_0)}) \times \Bbb{Z}/({p_1}^{\min(i_1, j_1)})$ and $\Bbb{Z}/(n) \simeq \Bbb{Z}/({p_0}^{\max(i_0, j_0)}) \times \Bbb{Z}/({p_1}^{\max(i_1, j_1)})$. This shows that $\Bbb{Z}/(a) \times \Bbb{Z}/(b)$ and $\Bbb{Z}/(m) \times \Bbb{Z}/(n)$ are (up to isomorphism) just rearrangements of the same factors, so again we're done.
Finally we extend this argument to any number of primes $p_k$.
