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 4.4 - Page 190: 42

Answer

See below.

Work Step by Step

Based on the quotient-remainder theorem, any integer $n$ must be in one of the following forms: n mod 6=0,1,2,3,4,5, or $n=6q, 6q+1,6q+2,6q+3,6q+4,6q+5$. Instead of proving $6q+1$ or $6q+5$ are prime numbers, we only need to prove that others must not be prime numbers. Case1: $n=6q$ has factors 2 and 3, thus it can not be a prime number; Case2: $n=6q+2$ has a factors 2, thus it can not be a prime number; Case3: $n=6q+3$ has a factors 3, thus it can not be a prime number; Case4: $n=6q+4$ has a factors 2, thus it can not be a prime number; Based on the above results, all prime numbers (except 2 and 3) must be in one of the forms of $6q+1$ or $6q+5$.
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