Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 4 - Section 4.1 - Divisibility and Modular Arithmetic - Exercises - Page 244: 6

Answer

If $a, b, c, d\in \mathbb{Z}$ with $a\neq 0$ s.t. $a|c$ and $b|d$, then $ab|cd$.

Work Step by Step

Let $a,b,c, d\in \mathbb{Z}$ with $a\neq 0$ s.t. $a|c$ and $b|d$. Because $a$ factors $c$, there exists an integer $g$ such that $a\cdot g=c$, and because $b$ factors $d$, there exists an integer $h$ such that $b\cdot h=d$. If $ab$ is a factor of $cd$, then there exists some integer $l$ where $ab\cdot l=cd$. Combining our two equations, $(a\cdot g)(b\cdot h)=c\cdot d=ab\cdot (gh)=cd$. $gh$ is the integer satisfying the condition for $l$, so this proves the statement above.
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