Basic divisibility and linear combinations

1. If $d\mid a$ and $d\mid b$, there exist integers $k,\ell$ with $a=dk$, $b=d\ell$. Then
$$a+b=d(k+\ell),\qquad a-b=d(k-\ell),$$
so $d\mid a+b$ and $d\mid a-b$.

2. With the same notation, $ax+by=dkx+d\ell y=d(kx+\ell y)$, so $d\mid ax+by$.

3. $60=2^2\cdot3\cdot5$. The positive divisors are
$$1,2,3,4,5,6,10,12,15,20,30,60.$$
There are $(2+1)(1+1)(1+1)=12$ of them.

Remark. Part (2) is the fundamental tool for all divisibility problems: any common divisor of $a$ and $b$ divides every integer linear combination of $a$ and $b$.