LCM from Prime Decomposition/Examples/125 and 150

Example of Use of LCM from Prime Decomposition

The lowest common multiple of $125$ and $150$ is:

$\lcm \set {125, 150} = 750$

Proof

\(\ds 125\)

\(=\)

\(\ds 5^3\)

\(\ds 150\)

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds 125\)

\(=\)

\(\ds 2^0 \times 3^0 \times 5^3\)

\(\ds 150\)

\(=\)

\(\ds 2^1 \times 3^1 \times 5^2\)

\(\ds \leadsto \ \ \)

\(\ds \lcm \set {125, 150}\)

\(=\)

\(\ds 2^1 \times 3^1 \times 5^3\)

\(\ds \)

\(=\)

\(\ds 750\)

$\blacksquare$

Sources

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