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

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $12$