Lowest Common Multiple of Integers/Examples/3054 and 12378

Example of Lowest Common Multiple of Integers

The lowest common multiple of $3054$ and $12378$ is:

$\lcm \set {3054, 12378} = 6 \, 300\, 402$

Proof

From Euclidean Algorithm: $12378$ and $3054$:

$\gcd \set {3054, 12378} = 6$

Then:

\(\ds \lcm \set {3054, 12378}\)

\(=\)

\(\ds \dfrac {3054 \times 12378} {\gcd \set {3054, 12378} }\)

Product of GCD and LCM

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2 \times 3 \times 509} \times \paren {2 \times 3 \times 2063} } 6\)

\(\ds \)

\(=\)

\(\ds 2 \times 3 \times 509 \times 2063\)

\(\ds \)

\(=\)

\(\ds 6 \, 300\, 402\)

$\blacksquare$

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm