Characterization of least common multiple

Question

There is a nice (duality) characterization of the greatest common divisor (wiki) of two integers $a$ and $b$ known as the Bézout's identity: $gcd(a,b)$ is the smallest positive integer that can be written as $ax + by$.

Is there a similar characterization of the least common multiple of two integers $a$ and $b$ such as "$lcm(a,b)$ is the largest $\ldots$"?

Answer 1

You can use Bézout's identity and the following relationship between the greatest common divisor and the least common multiple to come up with a characterization of $lcm(a,b)$. $$gcd(a,b) \times lcm(a,b) = a\times b \quad... \;\;(1)$$ $$lcm(a,b) = \frac{ab}{ax+by}$$

However, I am not aware of any other characterizations.

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EDIT:

Added proof of (1)

Answer 2

Actually the definition of $\gcd(a,b)$, valid in any P.I.D., is that it is a generator of the ideal $(a,b)$. In $\mathbf Z$, there are two generators, but there is a canonical way to choose one: it is to take the positive generator. In $F[X]$ ($F$ a field), we usually choose the monic generator.

In other P.I.D.s, there is usually no canonical way, as the g.c.d. is defined within a unit factor. For instance, in ring of Gaussian integers $\mathbf Z[i]$, the units are $\;\{1,-1,i,-i\}$ and there is no specific reason to choose one particular g.c.d.

The dual characterization of the l.c.m. of $a$ and $b$ is that it is a generator of the ideal $\;(a)\cap (b)$. In $\mathbf Z$, we likewise choose the positive generator.