Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.3 - Page 178: 38

Answer

(a) $p_1^{2e_1}p_2^{2e_2}\cdots p_k^{2e_k}$ (b) $42, (5880)^2$ (c) $231, (4158)^2$

Work Step by Step

(a) Given $a=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, we have $a^2=p_1^{2e_1}p_2^{2e_2}\cdots p_k^{2e_k}$ (b) To find the least positive integer $n$, let $2^5\cdot 3\cdot 5^2\cdot 7^3\cdot n=2^6\cdot 3^2\cdot 5^2\cdot 7^4$, we have: $n=2\cdot 3\cdot 7=42$ and $2^5\cdot 3\cdot 5^2\cdot 7^3\cdot n=(2^3\cdot 3\cdot 5\cdot 7^2)^2=(5880)^2$ (c) To find the least positive integer $m$, let $2^2\cdot 3^5\cdot 7\cdot 11\cdot m=2^2\cdot 3^6\cdot 7^2\cdot 11^2$, we have: $m=3\cdot 7\cdot 11=231$ and $2^2\cdot 3^5\cdot 7\cdot 11\cdot m=(2\cdot 3^3\cdot 7\cdot 11)^2=(4158)^2$
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