Exercise 1.2.29 (Solvability of $(x,y) = g,\ [x,y] = l$ )

Question

Let $g$ and $l$ be given positive integers. Prove that integers $x$ and $y$ exist satisfying $(x,y) = g$ and $[x,y] = l$ if and only if $g \mid l$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $g$ and $l$ be given positive integers.
\begin{itemize}

\item

If $x \wedge y = g$ and $x \vee y = l$, then $x \wedge y \mid x$, and $x \mid x \vee y$, thus $g \mid l$.

\item

Conversely, suppose that $g \mid l$. Take $x = g,\ y = l$. By Problem 18, since $g \mid l$,
\begin{align*}
 x \wedge y &= g \wedge l = g,\
 x \vee y &= g \vee l = l.
 \end{align*}

So integers $x$ and $y$ exist satisfying $(x,y) = g$ and $[x,y] = l$.
\end{itemize}
\end{proof}

Exercise 1.2.29 (Solvability of $(x,y) = g,\ [x,y] = l$ )

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Exercise 1.2.29 (Solvability of $(x,y) = g,\ [x,y] = l$ )

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