Exercise 1.2.18 (gcd$(a,b)$ if $a$ divides $b$)

Question

Find the values of $(a,b)$ and $[a,b]$ if $a$ and $b$ are positive integers such that $a \mid b$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

\begin{proof} By definition, if $a,b,d \in \Z$, then

$$a \wedge b = d \iff 
\left\{
\begin{array}{l}
(i) \quad d\geq 0,\
(ii) \quad  d \mid a, d \mid b,\
(iii) \quad  \forall \delta \in \Z,\ (\delta \mid a, \delta \mid b) \Rightarrow \delta \mid d. 
\end{array}
ight.
$$
\begin{enumerate}
\item[(a)]
If $a \mid b$, where $a\geq 0, b  \geq 0$, we show that
$$a \wedge b = a.$$
Indeed,
\begin{enumerate}
\item[(i)] $a\geq 0$,
\item[(ii)]  $a \mid a$ and $a \mid b$,
\item[(ii)] If $\delta\mid a$ and $\delta \mid b$, then $\delta \mid a$. $\qquad ((P 	ext{ and } Q) \Rightarrow P \ !)$ 
\end{enumerate}
\item[(b)] We show
$$a \vee b = b.$$
\begin{enumerate}
\item[(i)] $b\geq 0$,
\item[(ii)]  $a \mid b$ and $b \mid b$,
\item[(ii)] If $a \mid \mu $ and $b \mid \mu$, then $b \mid \mu$. 
\end{enumerate}
\end{enumerate}
\end{proof}

Note: in general, if $a \mid b$ then $a \wedge b = |a|$, and $a \vee b = |b|$.

Exercise 1.2.18 (gcd$(a,b)$ if $a$ divides $b$)

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Exercise 1.2.18 (gcd$(a,b)$ if $a$ divides $b$)

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