HMMT 二月 2009 · 冲刺赛 · 第 29 题

HMMT February 2009 — Guts Round — Problem 29

[ 15 ] The average of a set of distinct primes is 27. What is the largest prime that can be in this set?

解析

[ 15 ] The average of a set of distinct primes is 27. What is the largest prime that can be in this set? Answer: 139 Solution: Denote the set of these primes by A and the number of elements in A by n . There are 9 primes smaller than 27, namely 2,3,5,7,11,13,17,19 and 23. Since 27 is odd and all primes except 2 are odd, 2 ∈ / A . Thus the largest prime p is at most 27 · 9 − 3 − 5 − 7 − 11 − 13 − 17 − 19 − 23 = 145, so p ≤ 141. When the primes are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and 139, their average is 27. Therefore p = 139.