\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \DeclareMathOperator{\lcm}{lcm} \newcommand{\land}{\wedge} \newcommand{\lor}{\vee} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \newcommand{\fillinmath}[1]{\mathchoice{\underline{\displaystyle \phantom{\ \,#1\ \,}}}{\underline{\textstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptscriptstyle\phantom{\ \,#1\ \,}}}} \)

MATH/COMP SCI 240

Introduction to Discrete Mathematics

Spring 2025

The Division Algorithm

Multiples and Factors

Multiple
Divisor

Example 1.

\(3\mid 6\) since
\(5\mid 15\) since
\(2\nmid 7\) since

Activity

Are the following true or false?

\(\displaystyle 2\mid 0\)
\(\displaystyle 3\mid 11\)
\(\displaystyle 5\mid (20 + 15)\)
\(\displaystyle 20\mid 4\)
\(\displaystyle -3\mid 9\)

Linear Combinations

Theorem 2.

Let \(x\text{,}\) \(y\text{,}\) and \(z\) be integers. If \(x\mid y\) and \(x\mid z\text{,}\) then \(\fillinmath{XXXXXXXX}\) for any integers \(s\) and \(t\text{.}\)

Example 3.

\(3\mid 6\) and \(3\mid 9\text{.}\) What about \(6+9\) and \(2\cdot 6+3\cdot 9\text{?}\)