What is gcd?

It stands for greatest common divisor. It has two equivalent characterizations:

d = gcd(x, y) iff d is the largest thing such that d|x and d|y.

d = gcd(x, y) iff d is the smallest nonzero thing of the form ux + vy. (u and v need not be greater than zero)


(Size is measured by absolute value. We always use the positive one)


Note that all of this makes sense for more than just integers -- for example, it works for polynomials if "size" is measured by degree. (We always choose the monic polynomial)

 

I perfer a definition that does not require an ordering of element such as

gcd(x,y)=d iff d|x,y and if c|x,y then c|d
That is to say d is a common divisor of x and y
and
all common divisors of x and y divide d

 

Well, to be entirely silly, | is a (pre)ordering. (An actual partial ordering on things like the positive integers, or monic polynomials!)


I feel that the characterization as the least nonzero linear combination to be a generally more useful characterization, but I guess that gets translated in the same way: d = ux + vy, and d | ax + by for all a and b.