I've recently gotten in number theory, using Theory of Numbers by Andrew Adler as a starting point and came across a theorem that states,
Suppose a and b are not 0, let d = (a, b). Then d is the smallest positive integer that can be expressed as a linear combination of a and b.
Is the converse true, i.e. if I find an integer that is the smallest positive integer which is a linear combination of two integers a and b, then it must be the greatest common divisor of a and b? For example, if I found out 1 = xa + yb for some integer x and b, must 1 be the greatest common divisor, since it is the smallest positive integer possible?