Section 7.1 183 (b) Writing n as the product of prime powers as in part (a), the product (al + 1)(a2 + 1)··· (as + 1) is odd if and only if each factor ai + 1 is odd, hence, if and only if each ai = 2b i is even. And this occurs if and only if n = (p~lp~2 ... p~8)2 is a perfect square. (c) At the end of the day, a picket is white if and only if it has a number which is a perfect square. To see why, we need to determine those pickets whose color changes an even number of times. Picket k changes color each time there is an integer d, 1 < d S; k, which divides k. The number of such d, together with 1, is the total number of factors of d. By (a), this number is odd (and the number of d > 1 is even) if and only if k is a perfect square. Exercises 7.1 1. [BB] 13 . 12 . 11 . . ·6 = P(13, 8) = 51,891,840 2. 8· 7· 6 = P(8, 3) = 336 3. [BB] 10·9·8·7 = P(1O,4) = 5040 4. In 30 . 29 ... 23 = P(30, 8) ways. 5. [BB] 3 x 5!
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