LCM of 3 Integers in terms of GCDs of Pairs of those Integers

Theorem

Let $a, b, c \in \Z_{>0}$ be strictly positive integers.

Then:

$\lcm \set {a, b, c} = \dfrac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$

where:

$\gcd$ denotes greatest common divisor
$\lcm$ denotes lowest common multiple
$d_1 = \gcd \set {a, b}$
$d_2 = \gcd \set {b, c}$
$d_3 = \gcd \set {a, c}$

Lemma

Let $a, b, c \in \Z_{>0}$ be strictly positive integers.

Then:

$\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } = \gcd \set {a, b, c}$

Proof

 $\ds \lcm \set {a, b, c}$ $=$ $\ds \lcm \set {a, \lcm \set {b, c} }$ $\ds$ $=$ $\ds \frac {a \lcm \set {b, c} } {\gcd \set {a, \lcm \set {b, c} } }$ Product of GCD and LCM $\ds$ $=$ $\ds \frac {a b c} {\gcd \set {b, c} } \paren {\frac 1 {\gcd \set {a, \lcm \set {b, c} } } }$ Product of GCD and LCM $\ds$ $=$ $\ds \frac {a b c} {\gcd \set {b, c} } \paren {\frac 1 {\lcm \set {\gcd \set {a, b}, \gcd \set {a, c} } } }$ GCD and LCM Distribute Over Each Other $\ds$ $=$ $\ds \frac {a b c} {\gcd \set {b, c} } \paren {\frac {\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } } {\gcd \set {a, b} \gcd \set {a, c} } }$ Product of GCD and LCM $\ds$ $=$ $\ds \frac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$ by Lemma

$\blacksquare$