HMMT 十一月 2011 · 团队赛 · 第 1 题

HMMT November 2011 — Team Round — Problem 1

[ 2 ] Find the number of positive integers x less than 100 for which x x x x x x x 3 + 5 + 7 + 11 + 13 + 17 + 19 is prime. 3 2

解析

[ 2 ] Find the number of positive integers x less than 100 for which x x x x x x x 3 + 5 + 7 + 11 + 13 + 17 + 19 is prime. Answer: 0 We claim that our integer is divisible by 3 for all positive integers x . Indeed, we have x x x x x x x x x x x x x x 3 + 5 + 7 + 11 + 13 + 17 + 19 ≡ (0) + ( − 1) + (1) + ( − 1) + (1) + ( − 1) + (1) x x ≡ 3[(1) + ( − 1) ] ≡ 0 (mod 3) . It is clear that for all x ≥ 1, our integer is strictly greater than 3, so it will always be composite, making our answer 0. 3 2