Exercise 1.2.16 (gcd $=$ lcm)

Question

Prove that if $a$ and $b$ are positive integers satisfying $(a,b) = [a,b]$, then $a= b$.

Richard Ganaye · Accepted Answer

I write $a \wedge b = \mathrm{gcd}(a,b)$, and $a \vee b = \mathrm{lcm}(a,b)$.

\begin{proof} Assume that $a \wedge b = a \vee b$.

Then  $a \mid a \vee b$, and $a \vee b = a \wedge b$, thus $a \mid a \wedge b$, where $a \wedge b \mid b$, therefore $a \mid b$.

Exchanging the roles of $a$ and $b$, we obtain $b \mid a$.

Since $a\mid b$ and $b\mid a$, where $a>0,b>0$, then $a= b$.
\end{proof}

Answers