LCM of Coprime Integers

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Theorem

Let $a, b \in \Z_{>0}$ be coprime integers.


Then:

$\lcm \set {a, b} = a b$

where $\lcm$ denotes the lowest common multiple.


Proof

\(\displaystyle \lcm \set {a, b}\) \(=\) \(\displaystyle \frac {a b} {\gcd \set {a, b} }\) Product of GCD and LCM
\(\displaystyle \) \(=\) \(\displaystyle \frac {a b} 1\) Definition of Coprime Integers
\(\displaystyle \) \(=\) \(\displaystyle a b\)

$\blacksquare$