Lowest Common Multiple of Integers/Examples/27 and 81

Example of Lowest Common Multiple of Integers

The lowest common multiple of $27$ and $81$ is:

$\lcm \set {27, 81} = 81$

Proof

We find the greatest common divisor of $27$ and $81$ using the Euclidean Algorithm:

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

\(\ds 81\)

\(=\)

\(\ds 3 \times 27\)

Thus $\gcd \set {27, 81} = 27$.

Then:

\(\ds \lcm \set {27, 81}\)

\(=\)

\(\ds \dfrac {27 \times 81} {\gcd \set {27, 81} }\)

Product of GCD and LCM

\(\ds \)

\(=\)

\(\ds \dfrac {27 \times 81} {27}\)

\(\ds \)

\(=\)

\(\ds 81\)

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $5 \ \text {(c)}$