Lowest Common Multiple of Integers/Examples/25 and 30

Example of Lowest Common Multiple of Integers

The lowest common multiple of $25$ and $30$ is:

$\lcm \set {25, 30} = 150$

Proof

We find the greatest common divisor of $25$ and $30$ using the Euclidean Algorithm:

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

\(\ds 30\)

\(=\)

\(\ds 1 \times 25 + 5\)

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

\(\ds 25\)

\(=\)

\(\ds 5 \times 5\)

Thus $\gcd \set {25, 30} = 5$.

Then:

\(\ds \lcm \set {25, 30}\)

\(=\)

\(\ds \dfrac {25 \times 30} {\gcd \set {25, 30} }\)

Product of GCD and LCM

\(\ds \)

\(=\)

\(\ds \dfrac {5^2 \times 5 \times 6} 5\)

\(\ds \)

\(=\)

\(\ds 5^2 \times 6\)

\(\ds \)

\(=\)

\(\ds 150\)

$\blacksquare$

Sources

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