The reasoning behind why $7\mathbb{Z}_{28}$ is a summand of $\mathbb{Z}_{28}$

Question

So, apparently it comes down to the fact that $7\mathbb{Z}_{28}+ 4\mathbb{Z}_{28}= \mathbb{Z}_{28}$. But, how should one know in that these two groups would work. I'm asking because the numbers could be changed and I want to know the general procedure for such questions.

Answer 1

Whenever $m$ and $n$ are positive integers with $\gcd(m,n)=1$, then $$\Bbb Z_{mn}=m\Bbb Z_{mn}\oplus n\Bbb Z_{mn}.\tag{*}$$ This boils down to Bezout's identity; there are integers $a$ and $b$ with $am+bn=1$. This implies $1\in m\Bbb Z_{mn}+ n\Bbb Z_{mn}$ and so $\Bbb Z_{mn}= m\Bbb Z_{mn}+n\Bbb Z_{mn}$. But $|m\Bbb Z_{mn}|=n$ and $|n\Bbb Z_{mn}|=m$ so the sum is direct.