gcd(a,b) divides all linear combos of a and b.
Suppose and are integers - not both zero, and let be the set of all linear combinations of and .
The following relationship holds: Every linear combination of and (a member in ) is a multiple of ; conversely, any multiple of is a linear combination of and .
In short, is precisely the set of all multiples of .
BTW, the greatest common divisor of and is the smallest postive linear combination of and .