Discrete Mathematics with Graph Theory (3rd Edition) 79

# Discrete Mathematics with Graph Theory (3rd Edition) 79 -...

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Section 4.2 n 8. [BB] We have a = qln + r and b = q2n + r for some integers ql, q2 and r. Subtracting, a - b = (qln + r) - (q2n + r) = {ql - q2)n. Thus, n 1 (a - b) as required. 9. [BB] (-) If lOa + b = 7k for some integer k, then 5{lOa + b) = 35k, so 49a + a + 5b = 35k and a + 5b = 35k - 49a is divisible by 7. (f-) Conversely, if a + 5b = 7k, then lO{a + 5b) = 70k, so lOa + 50b = lOa + b + 49b = 70k and lOa + b = 70k - 49b is divisible by 7. 10. (a) [BB] True. Since alb, b = ax for some integer x. Since b 1 (-c), -e = by for some integer y. Thus, e = -by = -axy = a{ -xy). Since -xy is an integer, ale. (b) False. Let a = 3, b = 15, e = 3. Then 3115, so alb and e 1 b, but ae = 9 ~ 15. (c) True. Since alb, b = ax for some integer x.Thus, be = axe = a{xe). Since xc is an integer, a 1 be. (d) True. Since alb, b = ax for some integer x. Since e 1 d, d = cy for some integer y. Thus, bd = axcy = ae{ xy). Since xy is an integer, ae 1 bd. (e) True. Since
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