> I will willingly supply more references if you need them.
I don't. :-) I've taught more elementary number classes and reviewed more elementary number theory texts (including Rosen's) than I care to remember, and I have plenty of my own references. I stand by my assertion that the fractions module gcd is not wrong: it returns 'a' greatest common divisor for arbitrary integer inputs.
A bit more: the concept of greatest common divisor is slightly ambiguous: you can define the notion of "a greatest common divisor" for an arbitrary commutative ring-with-a-1 R: c is a greatest common divisor of a and b if c is a common divisor (i.e. c divides a and c divides b, where "x divides y" is synonymous with "y is a multiple of x"), and any other common divisor divides c. No ordering is necessary: "greatest" here is with respect to the divisibility lattice rather than with respect to any kind of total ordering. One advantage of this definition is that it makes it clear that 0 is a greatest common divisor of 0 and 0.
If further R is an integral domain, then it follows immediately from the definition that any two greatest common divisors of a and b (if they exist) are associates: a is a unit times b. In the particular case where R is the usual ring of rational integers, that means that "the" greatest common divisor of two numbers a and b is only really defined up to +/-; that is, the sign of the result is unimportant. (An alternative viewpoint is to think of the gcd, when it exists, as a principal ideal rather than an element of the ring.)
See https://proofwiki.org/wiki/Definition:Greatest_Common_Divisor/Integral_Domain for more along these lines.
So you're using one definition, I'm using another. Like I said, there's no universal agreement. ;-). |