Discrete Mathematics with Graph Theory (3rd Edition) 171

Discrete Mathematics with Graph Theory (3rd Edition) 171 -...

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Section 6.1 169 and so the number of integers relatively prime to 9975 is the number of integers not in A U B U CUD; that is, 9975 - 5655 = 4320. 13. [BB] Let A, B, and C be the sets of integers between 1 and 250 which are divisible by 4, by 6, and by 15, respectively. We want IA U B U 01. Now IAI = l2~0 J = 62, IBI = l2~0 J = 41, ICI = l2 1 5 5 0 J = 16 IA n BI = l2 1 5 2 0 J = 20, IA n 01 = l265~ J = 4, IB n 01 = l235~ J = 8 IA n B n CI = l250/60J = 4. SO IA U B U 01 = 62 + 41 + 16 - 20 - 4 - 8 + 4 = 91. 14. (a) [BB] Let A, B, C, D be the sets of integers between 1 and 1000 (inclusive) which are divisible by 2, by 3, by 5, and by 7, respectively. We want I(A U B U C U D)CI. Now IAUBuCuDI = IAI + IBI + ICI + IDI We have - IA n BI - IA n 01- IA n DI -IB n ci -IB n DI - IC n DI + IA n B n 01 + IA n B n DI + IAncnDI + IBncnDI-IAnBncnDI· IAI = ll0200 J = 500, IBI = ll0300 J = 333, ICI = llO~O J = 200, IDI = llO~O J = 142, IA n BI = llO~O J = 166, IA n CI = ll~gO J = 100, IA n DI
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Unformatted text preview: J = 71, IB n ci = ll~~O J = 66, IB n DI = llg~O J = 47, IC n DI = ll~~O J = 28, IA n B n 01 = ll~gO J = 33, IA n B n DI = ll~gO J = 23, IA n C n DI = ll~gO J = 14, IB n C n DI = l \~050 J = 9, IA n B n C n DI = l1201000 J = 4. SO IA U B U C U DI = 772. Our answer is 1000 - 772 = 228. (b) Now 1 and 1000 must be excluded. Since 1 is not divisible by 2, 3, 4 or 7 while 1000 is divisible by 5, the answer is 228 - 1 = 227. 15. Let A, B, C, D be the sets of integers between 1 and 10,000 which are divisible by 3, by 5, by 7, and by 11, respectively. We have IAI = llO~OO J = 3333, IBI = llO~OO J = 2000, ICI = llO~OO J = 1428, IDI = llOlOlOO J = 909, IA n BI = l10 1 5 00 J = 666, IA n 01 = llO~lOO J = 476, IA n DI = ll030300 J = 303, IB n CI = ll030500 J = 285, IB n DI = llO~500 J = 181, IC n DI = llO~~O J = 129, IA n B n CI = ll~g~O J = 95, IA n B n DI = ll~~~O J = 60, IA n C n DI = llg~~O J = 43, IB n C n DI = ll~~~O J = 25, IA n B n C n DI = l \155 J = 8....
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