LCM from Prime Decomposition/Examples/p^2 q and p q r
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Example of Use of LCM from Prime Decomposition
The lowest common multiple of $p^2 q$ and $p q r$, where $p$, $q$ and $r$ are all primes, is:
- $\lcm \set {p^2 q, p q r} = p^2 q r$
Proof
\(\ds p^2 q\) | \(=\) | \(\ds p^2 q^1 r^0\) | ||||||||||||
\(\ds p q r\) | \(=\) | \(\ds p^1 q^1 r^1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lcm \set {p^2 q, p q r}\) | \(=\) | \(\ds p^2 q^1 r^1\) | |||||||||||
\(\ds \) | \(=\) | \(\ds p^2 q r\) |
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $9 \ \text {(f)}$