How do you define least common multiple in the structure $(\mathbb{Z}, | , +, 0, 1)$

Question

Let the relation $\text{lcm$(x,y,z)$}$ have the meaning '$|z|$ is the least common multiple of $|x|$ and $|y|$'. Show $\text{lcm$(x,y,z)$}$ is definable in $(\mathbb{Z}, | , +, 0, 1)$.

My instinct is to do something like the following:

$\text{lcm$(x,y,z)$}\iff x|z \ \land \ y|z \ \land \forall w (w<z \implies (\neg( x|w) \lor \neg(y|w) ))$

but this is incorrect because as far as I can tell there's no way to define ordering in this structure.

Answer 1

A least common multiple of integers $a$ and $b$ is an integer $m$ such that

• $m$ is a common multiple of $a$ and $b$—that is, $a \mid m$ and $b \mid m$; and
• $m$ divides all other common multiples of $a$ and $b$—that is, for all $c \in \mathbb{Z}$, if $a \mid c$ and $b \mid c$, then $m \mid c$.

In other words, the least common multiple is 'least' in the sense of the divisibility relation $|$, rather than the order relation $\le$ (or $<$).

With this definition, least common multiples are unique up to sign.

But that's exactly why the absolute value signs have been included in your definition!

Answer 2

The important thing about least common multiple (and greatest common divisor, for that matter) is that they can both be defined in terms of the partial ordering given by divisibility. $lcm(n,m)$ is the common multiple of $n$ and $m$ such that every other common multiple of $n$ and $m$ is a multiple of $lcm(n,m)$.

The proof of this (and/or the similar statement for gcd) is usually covered in a first course of number theory. It might be worth reviewing and solidifying your understanding of that material, so you can concentrate on the logical stuff you're being introduced to.