HMMT 二月 2026 · ALGNT 赛 · 第 5 题
HMMT February 2026 — ALGNT Round — Problem 5
题目详情
- Compute the largest positive integer n such that √ n ! n divides ( ⌊ n ⌋ )! + 450 .
解析
- Compute the largest positive integer n such that √ n ! n divides ( ⌊ n ⌋ )! + 450 . Proposed by: Pitchayut Saengrungkongka Answer: 1230 √ n ! Solution: We will characterize all positive integer n such that n divides ( ⌊ n ⌋ )! + k , where k is a fixed positive integer. √ n ! Claim 1. We have n divides ( ⌊ n ⌋ )! + k if and only if one of the following holds: • n | k ; or • n = dp , where d | k and p is a prime such that p > d and p | k + 1 . √ n ! Proof. Let n be such that n | ( ⌊ n ⌋ )! + k and consider any prime divisor q of n . • If q | k , then we note that √ n ! ν (( ⌊ n ⌋ )! ) ≥ n ! > ν ( n ) , q q so we must have that ν ( n ) ≤ ν ( k ) . q q √ √ • If q does not divide k , then q does not divide ( ⌊ n ⌋ )! , so q > ⌊ n ⌋ . In particular, there exists at most one such q . In particular, we have two cases. • If every prime dividing n also divides k , then from the first bullet point, n | k . √ • If there exists a prime p such that p | n but p ∤ k , then p > ⌊ n ⌋ . Let n = pd . Then d < p . Moreover, from the first bullet point, ν ( d ) ≤ ν ( k ) for all prime q . Thus, d | k . Finally, by q q √ √ p − 1 n ! Fermat’s little theorem, ( ⌊ n ⌋ ) ≡ 1 (mod p ) , so ( ⌊ n ⌋ ) ≡ 1 (mod p ) , which implies that p | k + 1 . Finally, we extract the answer. We can easily check that 1230 = 30 · 41 works because 30 | 450 , 41 | 451 , and 30 < 41 . We now have to show that this is the largest possible solution. • If n | 450 , then n ≤ 450 . • If n = kp where k | 450 , p | 451 , and k ≤ p , then from p | 451 = 11 · 41 , we have that p is either 11 or 41 . – If p = 11 , then n ≤ 121 . – If p = 41 , then we have to list all divisors of 450 that are less than 41 . The largest one is 30 , so n ≤ 30 · 41 = 1230 . Remark. For concreteness, the complete list of all n are • (divisors of 450 ) 1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 25 , 30 , 45 , 50 , 75 , 90 , 150 , 225 , 450 , • (multiples of 11 ) 11 , 22 , 33 , 55 , 66 , 99 , 110 . • (multiples of 41 ) 41 , 82 , 123 , 205 , 369 , 410 , 615 , 738 , 1025 , 1230 . ©2026 HMMT