HMMT 二月 2003 · 代数 · 第 9 题

HMMT February 2003 — Algebra — Problem 9

解析

For how many integers n , for 1 ≤ n ≤ 1000, is the number even? 2 n Solution: 990 ( ) 2 n In fact, the expression is always even, and it is not a multiple of four if and only n if n is a power of 2, and there are 10 powers of 2 between 1 and 1000. Let f ( N ) denote the number of factors of 2 in N . Thus, ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ∞ ∑ n n n n f ( n !) = + + + · · · = . k 2 4 8 2 k =1 a Also, it is clear that f ( ab ) = f ( a ) + f ( b ) and f ( ) = f ( a ) − f ( b ) for integers a, b . Now b m m +1 for any positive integer n , let m be the integer such that 2 ≤ n < 2 . Then (( )) ( ) ( ) ⌊ ⌋ ⌊ ⌋ ∞ ∞ ∑ ∑ 2 n (2 n )! 2 n n f = f = − 2 k k n n ! n ! 2 2 k =1 k =1 ( ) ⌊ ⌋ ⌊ ⌋ ∞ ∞ ∑ ∑ n n = − 2 k − 1 k 2 2 k =1 k =1 ( ) ⌊ ⌋ ∞ ∑ n = b n c − k 2 k =1 ( ) ⌊ ⌋ m ∑ n = n − k 2 k =1 ( ) m ∑ n ≥ n − k 2 k =1 ( ) m 2 − 1 n = n − n = ≥ 1 . m m 2 2 ( ) 2 n m Both equalities hold when n = 2 , and otherwise, f ( ) > 1. n