HMMT 十一月 2014 · 冲刺赛 · 第 25 题

HMMT November 2014 — Guts Round — Problem 25

[ 13 ] 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 . • n = pq for some distinct prime numbers p and q

解析

[ 13 ] 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 . • n = pq for some distinct prime numbers p and q k Answer: 40 The first numbers which are neither of the form p nor pq are 12 , 18 , 20 , 24 , 28 , 30 , 36 , 40 , . . . . Of these 12 , 18 , 20 , 24 , 30 are of the form p + 1 and 28, 36 are triangular. Hence the answer is 40.