Lowest Common Multiple of Integers/Examples/6 and 15

Example of Lowest Common Multiple of Integers

The lowest common multiple of $6$ and $15$ is:

$\lcm \set {6, 15} = 30$

Proof

We find the greatest common divisor of $6$ and $15$ using the Euclidean Algorithm:

\(\text {(1)}: \quad\)

\(\ds 15\)

\(=\)

\(\ds 2 \times 6 + 3\)

\(\text {(2)}: \quad\)

\(\ds 6\)

\(=\)

\(\ds 2 \times 3\)

Thus $\gcd \set {6, 15} = 3$.

Then:

\(\ds \lcm \set {6, 15}\)

\(=\)

\(\ds \dfrac {6 \times 15} {\gcd \set {6, 15} }\)

Product of GCD and LCM

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2 \times 3} \times \paren {3 \times 5} } 3\)

\(\ds \)

\(=\)

\(\ds 2 \times 3 \times 5\)

\(\ds \)

\(=\)

\(\ds 30\)

$\blacksquare$