least common multiple

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

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

is the positiveinteger $f$ 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 severalnumbers.  The positive lcm of positive integers isuniquely determined. (Its negative satisfies the same twoconditions.)

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 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 begeneralized for several numbers, i.e., for example,

   $$\gcd (a,b,c)\cdot \mathrm{lcm}(a,b,c)\ne abc.$$

3. The preceding formula may be presented in of ideals of $\mathbb{Z}$; we mayreplace 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

   $$(𝔞+𝔟)(𝔞\cap 𝔟)=𝔞𝔟$$

   is true for all ideals $𝔞$ and $𝔟$ of $R$, with at least one of them having non-zero-divisors (http://planetmath.org/ZeroDivisor).