LCM from Prime Decomposition/Examples/39 and 143

Example of Use of LCM from Prime Decomposition

The lowest common multiple of $39$ and $143$ is:

$\lcm \set {39, 143} = 429$

Proof

\(\ds 39\)

\(=\)

\(\ds 3 \times 13\)

\(\ds 143\)

\(=\)

\(\ds 11 \times 13\)

\(\ds \leadsto \ \ \)

\(\ds 39\)

\(=\)

\(\ds 3^1 \times 11^0 \times 13^1\)

\(\ds 143\)

\(=\)

\(\ds 3^0 \times 11^1 \times 13^1\)

\(\ds \leadsto \ \ \)

\(\ds \lcm \set {39, 143}\)

\(=\)

\(\ds 3^1 \times 11^1 \times 13^1\)

\(\ds \)

\(=\)

\(\ds 429\)

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $9 \ \text {(c)}$