hw1 - SOLUTIONS FOR HOMEWORK 1 Bonus problem: 1.25. Answer:...

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SOLUTIONS FOR HOMEWORK 1 Bonus problem: 1.25. Answer: 9 , 2 , 2 . To see this, denote the ages of the daughters, in the non-increasing order, by a 1 , a 2 , a 3 . We know that: (1) a 1 > a 2 a 3 (see the reference to “the eldest daughter”). (2) a 1 a 2 a 3 = 36. (3) Moreover, if b 1 > b 2 b 3 , a 1 + a 2 + a 3 = b 1 + b 2 + b 3 , and b 1 b 2 b 3 = 36, then a 1 = b 1 , a 2 = b 2 , and a 3 = b 3 (the census taker managed to compute the ages based on the information he had). (4) However, there exist b 1 = b 2 b 3 such that a 1 + a 2 + a 3 = b 1 + b 2 + b 3 , and b 1 b 2 b 3 = 36 (the extra bit of info about “the eldest daughter” was needed). Let’s look at all the factorizations a 1 a 2 a 3 = 36, with a 1 > a 2 a 3 : a 1 a 2 a 3 a 1 + a 2 + a 3 36 1 1 38 18 2 1 21 12 3 1 16 9 4 1 15 9 2 2 13 6 3 2 11 4 3 3 10 There is only one factorization b 1 b 2 b 3 = 36 with b 1 = b 2 a 3 : b 1 = b 2 = 6, and b 3 = 1. Then b 1 + b 2 + b 3 = 13. The number 13 occurs in the above table ponly
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hw1 - SOLUTIONS FOR HOMEWORK 1 Bonus problem: 1.25. Answer:...

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