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# 222a3s - MATH 222 FALL 2011 Assignment Three(Answer Key 1...

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Unformatted text preview: MATH 222 FALL 2011, Assignment Three (Answer Key) 1. Find the number of integers between 1 and 10,000 inclusive which are (a) divisible by at least one of 3, 5, 7, 11; For i = 3 , 5 , 7 , 11, let S i be the set of integers between 1 and 10,000 divisible by i . We need to compute | S 3 ∪ S 5 ∪ S 7 ∪ S 11 | , which is (by the P.I.E.) equal to | S 3 | + | S 5 | + | S 7 | + | S 11 |- | S 3 ∩ S 5 | + | S 3 ∩ S 7 | − | S 3 ∩ S 11 | − | S 5 ∩ S 7 | − | S 5 ∩ S 11 | − | S 7 ∩ S 11 | + | S 3 ∩ S 5 ∩ S 7 | + | S 3 ∩ S 5 ∩ S 11 | + | S 3 ∩ S 7 ∩ S 11 | + | S 5 ∩ S 7 ∩ S 11 |- | S 3 ∩ S 5 ∩ S 7 ∩ S 11 | where | S 3 | = ⌊ 10 , 000 / 3 ⌋ , | S 5 | = ⌊ 10 , 000 / 5 ⌋ , | S 7 | = ⌊ 10 , 000 / 7 ⌋ , | S 11 | = ⌊ 10 , 000 / 11 ⌋ , | S 3 ∩ S 5 | = ⌊ 10 , 000 / 15 ⌋ , | S 3 ∩ S 7 | = ⌊ 10 , 000 / 21 ⌋ , | S 3 ∩ S 11 | = ⌊ 10 , 000 / 33 ⌋ , | S 5 ∩ S 7 | = ⌊ 10 , 000 / 35 ⌋ , | S 5 ∩ S 11 | = ⌊ 10 , 000 / 55 ⌋ , | S 7 ∩ S 11 | = ⌊ 10 , 000 / 77 ⌋ , | S 3 ∩ S 5 ∩ S 7 | = ⌊ 10 , 000 / 105 ⌋ , | S 3 ∩ S 5 ∩ S 11 | = ⌊ 10 , 000 / 165 ⌋ , | S 3 ∩ S 7 ∩ S 11 | = ⌊ 10 , 000 / 231 ⌋ , | S 5 ∩ S 7 ∩ S 11 | = ⌊ 10 , 000 / 385 ⌋ , and | S 3 ∩ S 5 ∩ S 7 ∩ S 11 | = ⌊ 10 , 000 / 1155 ⌋ ....
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222a3s - MATH 222 FALL 2011 Assignment Three(Answer Key 1...

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