Least Common Multiple

The least common multiple of two numbers $n_1$ and $n_2$ is denoted $\mathop{\rm LCM}\nolimits (n_1, n_2)$ or $[n_1,n_2]$ and can be obtained by finding the Prime factorization of each

$$n_1={p_1}^{a_1}\cdots{p_n}^{a_n}$$ (1)

$$n_2={p_1}^{b_1}\cdots{p_n}^{b_n},$$ (2)

where the $p_i$s are all Prime Factors of $n_1$ and $n_2$, and if $p_i$ does not occur in one factorization, then the corresponding exponent is 0. The least common multiple is then

$$\mathop{\rm LCM}\nolimits (n_1,n_2) = [n_1,n_2] = \prod_{i=1}^n {p_i}^{\max(a_i,b_i)}.$$ (3)

Let $m$ be a common multiple of $a$ and $b$ so that

$$m=ha=kb.$$ (4)

Write $a=a_1(a,b)$ and $b=b_1(a,b)$, where $a_1$ and $b_1$ are Relatively Prime by definition of the Greatest Common Divisor $(a_1,b_1)=1$. Then $ha_1=kb_1$, and from the Division Lemma (given that $ha_1$ is Divisible by $b$ and $(b_1,a_1)=0$), we have $h$ is Divisible by $b_1$, so

$$h=nb_1$$ (5)

$$m=ha=nb_1a = n{ab\over(a,b)}.$$ (6)

The smallest $m$ is given by $n=1$,

$$\mathop{\rm LCM}\nolimits (a,b) = {ab\over \mathop{\rm GCD}\nolimits (a,b)},$$ (7)

so

$$\mathop{\rm GCD}\nolimits (a,b)\,\mathop{\rm LCM}\nolimits (a,b)=ab$$ (8)

$$(a,b)[a,b]=ab.$$ (9)

The LCM is Idempotent

$$[a,a]=a,$$ (10)

Commutative

$$[a,b]=[b,a],$$ (11)

Associative

$$[a,b,c]=[[a,b],c]=[a,[b,c]],$$ (12)

Distributive

$$[ma,mb,mc]=m[a,b,c],$$ (13)

and satisfies the Absorption Law

$$(a,[a,b])=a.$$ (14)

It is also true that

$$[ma,mb]={(ma)(mb)\over (ma,mb)} =m{ab\over (a,b)} = m[a,b].$$ (15)

See also Greatest Common Divisor, Mangoldt Function, Relatively Prime

References

Guy, R. K. ``Density of a Sequence with L.C.M. of Each Pair Less than $x$.'' §E2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 200-201, 1994.