Least common multiple and greatest common divisor for monomial with rational coefficients

Question

I have read in my textbook that if I have, for example two monomials $A$ e $B$ one or both with rational coefficients,

$$A=\frac 34 x^2y^3, \qquad B=-2xyz$$

for the $\text{lcm}$ or the $\text{gcd}$ we have always $1$ as numerical coefficient. Is it a convention or is there something more complex that I do not know?

Answer 1

$lcm$ or $gcd$ doesn't really come into play for the rational coefficient part, at least not in the precalc algebra that I know of.

Answer 2

It’s imprecise to speak of the least common multiple or the greatest common divisor in this context. A least common multiple is defined as a common multiple that divides all common multiples, and a greatest common divisor is defined as a common divisor that all common divisors divide. In a ring of polynomials with rational coefficients, polynomials that differ only by a constant non-zero factor divide each other, i.e. they’re associates. Thus, given a gcd/lcm, all its non-zero constant multiples are also gcds/lcms. Thus, the convention here is not that the associate with coefficient $1$ is the gcd/lcm; it’s just that the set of associates, all of which are gcds/lcms, is conventionally represented by the representative with coefficient $1$.