HMMT 十一月 2023 · 冲刺赛 · 第 21 题
HMMT November 2023 — Guts Round — Problem 21
题目详情
- [11] An integer n is chosen uniformly at random from the set { 1 , 2 , 3 , . . . , 2023! } . Compute the probability that n gcd( n + 50 , n + 1) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2023, November 11, 2023 — GUTS ROUND Organization Team Team ID#
解析
- [11] An integer n is chosen uniformly at random from the set { 1 , 2 , 3 , . . . , 2023! } . Compute the prob- ability that n gcd( n + 50 , n + 1) = 1 . Proposed by: Pitchayut Saengrungkongka 265 Answer: . 357 Solution: If n is even, we need gcd( n + 1 , 51) = 1. If n is odd, we need gcd( n + 1 , 49) = 1. Thus, the answer is 1 φ (49) φ (51) 265
- = . 2 49 51 357