Common multiple, and the least common multiple $\mathrm{lcm}(a,b)$, taken to be $0$ when $a=0$ or $b=0$

Let $a,b\in \mathbb{Z}$. An integer $m$ is a common multiple of $a$ and $b$ when $a\mid m$ and $b\mid m$ (Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$).

Case $a\mathrm{\ne }0$ and $b\mathrm{\ne }0$: a least positive common multiple exists. The integer $\mathrm{\mid }ab\mathrm{\mid }$ is a positive common multiple. It is a common multiple because $a\mid a$ gives $a\mid ab$ and hence $a\mid \mathrm{\mid }ab\mathrm{\mid }$, the last step because $\mathrm{\mid }ab\mathrm{\mid }$ is $ab$ or $-ab$ (Divisibility is reflexive and transitive on $\mathbb{Z}$, and is linear: if $d\mid a$ and $d\mid b$ then $d\mid ax+by$ for all integers $x,y$; also $d\mid a$ implies $d\mid ac$, $-d\mid a$ and $d\mid -a$, The absolute value $\mathrm{\mid }a\mathrm{\mid }$ of an integer), and symmetrically for $b$; and it is positive because $ab\mathrm{\ne }0$ (The integers have no zero divisors; multiplicative cancellation), so $\mathrm{\mid }ab\mathrm{\mid }\ge 0$ and $\mathrm{\mid }ab\mathrm{\mid }\mathrm{\ne }0$ (Absolute value in $\mathbb{Z}$: $\mathrm{\mid }a\mathrm{\mid }\ge 0$; $\mathrm{\mid }a\mathrm{\mid }=0$ exactly when $a=0$; $\mathrm{\mid }-a\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$; $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$; $-\mathrm{\mid }a\mathrm{\mid }\le a\le \mathrm{\mid }a\mathrm{\mid }$; and $\mathrm{\mid }a\mathrm{\mid }\le c$ exactly when $-c\le a\le c$). Every positive common multiple is nonnegative, hence of the form $\iota (k)$ for a unique $k\in \mathbb{N}$, where $\iota :\mathbb{N}\to \mathbb{Z}$ is the embedding of The naturals embed in the integers; so the set

$T:=\{k\in \mathbb{N}:\iota (k)\text{ is a positive common multiple of }a\text{ and }b\}$

is a nonempty subset of $\mathbb{N}$ and has a least element $k_0$ (The well-ordering principle). Since $\iota$ preserves the order, $\iota (k_0)$ is then the least positive common multiple of $a$ and $b$, and we define

$\mathrm{lcm}(a,b):=\iota (k_0),$

the least positive common multiple. It is unique, greatest and least elements being unique by antisymmetry (The integers form a totally ordered ring, Order on the integers).

Case $a=0$ or $b=0$: the only common multiple is $0$. Say $a=0$. Then $a\mid m$ reads $0\mid m$, which holds exactly for $m=0$ (Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$); and $0$ is indeed a common multiple, since every integer divides $0$. So there is no positive common multiple at all, and we set

$\mathrm{lcm}(a,b):=0.$

This is not a free choice dressed as one: $0$ is the only common multiple of the pair, so any other value would name an integer that is not a common multiple.

With both cases together $\mathrm{lcm}(a,b)$ is defined for every pair, and $\mathrm{lcm}(a,b)\ge 0$ always.

• The two boundary values agree, but they have different standing. $\gcd (0,0)=0$ is a genuine convention, adopted in Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$ because no greatest common divisor exists there; $\mathrm{lcm}(a,b)=0$ at a vanishing argument is not a convention at all, since $0$ is the only common multiple available. With the two together, $\gcd (a,b)\mathrm{lcm}(a,b)=\mathrm{\mid }ab\mathrm{\mid }$ holds at every pair without exception (Every common multiple of $a$ and $b$ is a multiple of $\mathrm{lcm}(a,b)$, and $\gcd (a,b)\cdot \mathrm{lcm}(a,b)=\mathrm{\mid }ab\mathrm{\mid }$). At $a=0$ that identity reads $\gcd (0,b)\cdot 0=0=\mathrm{\mid }0\cdot b\mathrm{\mid }$, which is true for every $b$.

• "Least" is least in the order of $\mathbb{Z}$, among the positive common multiples. The stronger statement, that $\mathrm{lcm}(a,b)$ divides every common multiple and not merely that it is smallest, is a theorem and is proved as the first half of Every common multiple of $a$ and $b$ is a multiple of $\mathrm{lcm}(a,b)$, and $\gcd (a,b)\cdot \mathrm{lcm}(a,b)=\mathrm{\mid }ab\mathrm{\mid }$. It is that divisibility form, not the minimality, that later pages use.

• Two arguments only. As with $\gcd$, no least common multiple of a longer list is defined on this page, because nothing here needs one.