According to the theorem, it says:

If $\displaystyle a \mid (b)$ and $\displaystyle a \mid (c)$, Then $\displaystyle a \mid mb + nc$ where $\displaystyle m,n \in \mathbb{Z}$

Does this mean $\displaystyle a \mid mb + nc$ where mb=divisor*quotient and nc=remainder?

Or does it mean $\displaystyle a \mid (mb + nc)$, where $\displaystyle a$ divides the whole thing?

Say if I have $\displaystyle m \mid (a-b)$ and $\displaystyle m \mid (c-d)$, I could say this is: $\displaystyle m \mid p(a-b)+q(c-d)$ where $\displaystyle p,q \in \mathbb{Z}$, is this right?

But if I do it the "long way" to get these:

$\displaystyle a=pm+b$

$\displaystyle c=qm+d$

$\displaystyle a+c=m(p+q)+b+d$ Here, the $\displaystyle (b+d)$ is the remainder, right? Sometimes it is hard to see if that it is part of the remainder or the quotient*divisor.

From $\displaystyle a+c=m(p+q)+b+d$, I convert to:

$\displaystyle a+c\equiv b+d \;\;\;\; (mod \; \;m)$

$\displaystyle m \mid a+c-(b+d)$

Here, I don't see $\displaystyle m \mid a+c-(b+d)$ similar to $\displaystyle m \mid p(a-b)+q(c-d)$... Am I not using the theorems correctly?