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Discrete Mathematics with Graph Theory (3rd Edition) 85

# Discrete Mathematics with Graph Theory (3rd Edition) 85 -...

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Section 4.3 83 am + bn = 1 implies am = -bn + 1, so (a + n)m = (-b + m)n + 1. Set a' = a + n and b' = -b + m. Because 0 < b < m, 0 < b' < m also. Because -n < a < 0, 0 < a' < n. The uniqueness of a' and b' follows from the uniqueness of a and b. (Uniqueness can also be proved directly as in (i).) Exercises 4.3 1. (a) [BB] 157 is prime;. (b) [BB] 9831 is not prime; 3 I 9831; (c) 9833 is prime; (d) 55,551,111 is not prime; 11155,551,111. (e) 2 216 ,090 - 1 is not prime: 2 216 ,090 - 1 = (2 108 ,045 + 1)(2 108 ,045 - 1). 2. No. For example, take n = 9. Then n has no prime.factor smaller than v'9. 3. (a) [BB] Note that n = p(nlp). Now if nip is not prime, then it has a prime factor q < Jnlp by Lemma 4.3.4. Since a prime factor of nip is a prime factor of n, q would then be a prime factor of n not exceeding Jnlp, contradicting the fact that the smallest prime factor of n is bigger than this. (b) [BB] Note that 16,773,121 = 433(38,737) and that .../38,737 ~ 197. Since 433, the smallest prime dividing 38,737, is larger than 197, 38,737 is prime, by part (a). Thus, 16,773,121 = 433(38,737) is the representation of 16,773,121 as the product of primes. 4. (a) [BB] 856 = 2 . 2 . 2 . 107 = 2 3 .
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