If $a$ and $b$ are two positive integers, then their least common multiple, denoted by

$\mathrm{lcm}\!(a,\,b),$ |

is the positive integer $f$ satisfying the conditions

- •
$a\mid f$ and $b\mid f$,

- •
if $a\mid c$ and $b\mid c$, then $f\mid c$.

Note: The definition can be generalized for several numbers. The positive lcm of positive integers is uniquely determined. (Its negative satisfies the same two conditions.)

## Properties

- 1.
If $a=\prod_{i=1}^{m}p_{i}^{\alpha_{i}}$ and $b=\prod_{i=1}^{m}p_{i}^{\beta_{i}}$ are the prime factor presentations of the positive integers $a$ and $b$ ($\alpha_{i}\geqq 0$, $\beta_{i}\geqq 0$ $\forall i$), then

$\mathrm{lcm}\!(a,\,b)=\prod_{i=1}^{m}p_{i}^{\max\{\alpha_{i},\,\beta_{i}\}}.$ This can be generalized for lcm of several numbers.

- 2.
Because the greatest common divisor has the expression $\gcd(a,\,b)=\prod_{i=1}^{m}p_{i}^{\min\{\alpha_{i},\,\beta_{i}\}}$, we see that

$\gcd(a,\,b)\cdot\mathrm{lcm}\!(a,\,b)=ab.$ This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,

$\gcd(a,\,b,\,c)\cdot\mathrm{lcm}(a,\,b,\,c)\neq abc.$ - 3.
The preceding formula may be presented in terms of ideals of $\mathbb{Z}$; we may replace the integers with the corresponding principal ideals. The formula acquires the form

$((a)+(b))((a)\cap(b))=(a)(b).$ - 4.
The recent formula is valid also for other than principal ideals and even in so general systems as the Prüfer rings; in fact, it could be taken as defining property of these rings: Let $R$ be a commutative ring with non-zero unity. $R$ is a Prüfer ring iff Jensen’s formula

$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b})=\mathfrak{ab}$ is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having non-zero-divisors.

## References

- 1 M. Larsen and P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).