Lowest Common Multiple of Integers/Examples/42 and 49

Example of Lowest Common Multiple of Integers

The lowest common multiple of $42$ and $49$ is:

$\lcm \set {42, 49} = 294$

Proof

We find the greatest common divisor of $42$ and $49$ using the Euclidean Algorithm:

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

\(\ds 49\)

\(=\)

\(\ds 1 \times 42 + 7\)

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

\(\ds 42\)

\(=\)

\(\ds 6 \times 7\)

Thus $\gcd \set {42, 49} = 7$.

Then:

\(\ds \lcm \set {42, 49}\)

\(=\)

\(\ds \dfrac {42 \times 49} {\gcd \set {42, 49} }\)

Product of GCD and LCM

\(\ds \)

\(=\)

\(\ds \dfrac {6 \times 7 \times 7^2} 7\)

\(\ds \)

\(=\)

\(\ds 6 \times 7^2\)

\(\ds \)

\(=\)

\(\ds 294\)

$\blacksquare$

Sources

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