Sunday, 2 November 2025

Exercise (2.4).9

Prove that $[a, ma] = ma$ where $m$ and $a$ are positive integers.


We use Proposition (2.22) that $\gcd (a, b) × [a, b] = a × b$.

So here we have

$$ \begin{align} [a,ma] &= \frac{a \times ma}{\gcd(a, ma)}\\ \\ &= \frac{a \times ma}{a \times \gcd(1, m)}\\ \\  &=  \frac{ma}{1} \quad \text{using } \gcd(1,m)=1 \\ \\ &= ma \end{align} $$