# Characterization of least common multiple

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 $\ldots$"?

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.

EDIT:

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.