divisibility - Properties of Integers

A theorem presented in my discrete math book.

Let $d$ be the smallest positive integer of the form $ax + by$.
Then $d = \gcd(a,b)$, where gcd means greatest common divisor.

I don't understand how the variable $d$ being the smallest possible integer from the expression ($ax + by$) results in the greatest common divisor.

It also doesn't state what are the allowed values of $a$, $b$, $x$, and $y$ are either.

My guess would be they want x and y to be integers.