Discrete Mathematics with Applications 4th Edition

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

Chapter 7 - Functions - Exercise Set 7.1 - Page 396: 51

Answer

$a-\phi(15)=8\\ b-\phi(2)=1\\ c-\phi(5)=4\\ d-\phi(12)=4\\ e-\phi(11)=10\\ f-\phi(1)=1\\$

Work Step by Step

$\phi(n) \,\,\,is\,defined\,as\,follows:\\ For\,each\,integer\,\,\,\, n \geq 1, \\\phi(n)\,is\,the\,number\,of\,positive\,integers\,less\,than\,or\,equal\,\, to\,n\,\,\\that\,have\,no\,common\,factors\,with\,\,n\,\,except\,\,\pm 1. \\ a-\phi(15)=8\\ because\,\,there\,are\,eight\,positive\,integers\,less\,than\,\,\\ or\,\,equal\,to\,15\,that\,have\,no\,common\,factors\,with\,15\,\,except\pm 1;\\ namely,1,2,4,7,8,11,13\,and\,14 \\ b-\phi(2)=1\\ because\,\,there\,are\,one\,positive\,integers\,less\,than\,\,\\ or\,\,equal\,to\,2\,that\,have\,no\,common\,factors\,with\,2\,\,except\pm 1;\\ namely,1\\ c-\phi(5)=4\\ because\,\,there\,are\,four\,positive\,integers\,less\,than\,\,\\ or\,\,equal\,to\,5\,that\,have\,no\,common\,factors\,with\,5\,\,except\pm 1;\\ namely,1,2,3,4\\ d-\phi(12)=4\\ because\,\,there\,are\,four\,positive\,integers\,less\,than\,\,\\ or\,\,equal\,to\,12\,that\,have\,no\,common\,factors\,with\,12\,\,except\pm 1;\\ namely,1,5,7,11\\ e-\phi(11)=10\\ because\,\,there\,are\,one\,ten\,integers\,less\,than\,\,\\ or\,\,equal\,to\,11\,that\,have\,no\,common\,factors\,with\,11\,\,except\pm 1;\\ namely,1,2,3,4,5,6,7,8,9,10\\ f-\phi(1)=1\\ because\,\,there\,are\,one\,positive\,integers\,less\,than\,\,\\ or\,\,equal\,to\,1\,that\,have\,no\,common\,factors\,with\,1\,\,except\pm 1;\\ namely,1\\ $
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