HMMT 十一月 2014 · 团队赛 · 第 1 题

HMMT November 2014 — Team Round — Problem 1

[ 3 ] What is the smallest positive integer n which cannot be written in any of the following forms? • n = 1 + 2 + · · · + k for a positive integer k . k • n = p for a prime number p and integer k . • n = p + 1 for a prime number p . 2

解析

[ 3 ] What is the smallest positive integer n which cannot be written in any of the following forms? • n = 1 + 2 + · · · + k for a positive integer k . k • n = p for a prime number p and integer k . • n = p + 1 for a prime number p . k Answer: 22 Consider 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , 17 , 19 are in the form p . So we are left with 6 , 10 , 12 , 14 , 15 , 18 , 20 , 21 , 22 , ... Next, 6 , 12 , 14 , 18 , 20 are in the form p + 1, so we are left with 10 , 15 , 21 , 22 , ... Finally, 10 , 15 , 21 are in the form n = 1 + 2 + · · · + k , so we are left with 22 , ... Since 22 = 2 · 11 is not a prime power, 22 − 1 = 21 is not prime, and 1 + 2 + · · · + 6 = 21 < 22 < 28 = 1 + 2 + · · · + 7, 22 is the smallest number not in the three forms, as desired. 2