PlanetMath: least common multiple

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least common multiple

(Definition)

If and are two positive integers, then their least common multiple, denoted by ${\mathrm{lcm}}(a,\,b)$ , is the positive integer 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 ${\mathrm{lcm}}$ of positive integers is uniquely determined. (Its negative satisfies the same two conditions.)

Properties

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 and ( $\alpha_{i} \geqq 0$ , $\beta_{i} \geqq 0$ $\forall i$ ), then
$\displaystyle {\mathrm{lcm}}(a,\,b)= \prod_{i=1}^{m}p_i^{\max\{\alpha_i,\,\beta_i\}}.$
This can be generalized for ${\mathrm{lcm}}$ of several numbers.
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
$\displaystyle \gcd(a,\,b)\cdot {\mathrm{lcm}}(a,\,b) = ab.$
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,
$\displaystyle \gcd(a,\,b,\,c)\cdot {\mathrm{lcm}}(a,\,b,\,c) \neq abc.$
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
$\displaystyle ((a)+(b))((a)\cap(b)) = (a)(b).$
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 be a commutative ring with non-zero unity. is a Prüfer ring iff Jensen's formula
$\displaystyle (\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$
is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of , with at least one of them having non-zero-divisors.