least common multiple

If $a$ and $b$ are two positive integers, then their least common multiple, denoted by $\lcm(a,\,b)$ , 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 $$\lcm(a,\,b)= \prod_{i=1}^{m}p_i^{\max\{\alpha_i,\,\beta_i\}}.$$ 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 $$\gcd(a,\,b)\cdot \lcm(a,\,b) = ab.$$ This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example, $$\gcd(a,\,b,\,c)\cdot \lcm(a,\,b,\,c) \neq abc.$$
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.

Bibliography

1

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