HMMT 二月 2026 · ALGNT 赛 · 第 3 题
HMMT February 2026 — ALGNT Round — Problem 3
题目详情
- Compute the sum of all positive integers n such that n has at least 6 positive integer divisors and the 6 th largest divisor of n is 6 .
解析
- Compute the sum of all positive integers n such that n has at least 6 positive integer divisors and the 6 th largest divisor of n is 6 . Proposed by: Srinivas Arun Answer: 48 Solution: Since 6 is a divisor of n , we know that 1 , 2 , and 3 are also divisors of n . We proceed by casework on which of 4 and 5 are divisors of n . • If neither 4 or 5 are divisors of n , then the four smallest divisors of n are 1 , 2 , 3 , and 6 , and the n n n four largest divisors of n are , , , and n . Moreover, since 6 is the 6 th largest divisor, we 6 3 2 n know that n has exactly one divisor between 6 and . This implies that n has an odd number of 6 divisors and is thus a perfect square, which is impossible because n is divisible by 2 but not 4 . ©2026 HMMT • If 4 is a divisor of n but 5 is not, then the five smallest divisors of n are 1 , 2 , 3 , 4 , and 6 , and n n n n the five largest divisors of n are , , , , and n . Moreover, since 6 is the 6 th largest divisor, 6 4 3 2 n we know that there are no divisors between 6 and . Therefore, n has exactly 10 divisors total. 6 4 Since n is divisible by 3 and 4 , we must have n = 2 · 3 = 48 , which is a solution. • If 5 is a divisor of n but 4 is not, then the five smallest divisors of n are 1 , 2 , 3 , 5 , and 6 , and n n n n the five largest divisors of n are , , , , and n . Moreover, since 6 is the 6 th largest divisor, 6 5 3 2 n we know that there are no divisors between 6 and . Therefore, n has exactly 10 divisors total, 6 which is impossible because n has at least three distinct prime factors ( 2 , 3 , and 5 ). • If both 4 and 5 are divisors of n , then the six smallest divisors of n are 1 , 2 , 3 , 4 , 5 , and 6 , and n n n n n the six largest divisors of n are , , , , , and n . Therefore, we must have n = 36 . However, 6 5 4 3 2 we assumed that n is divisible by 5 in this case, which leads to a contradiction. Finally, since the only answer is n = 48 , the final answer is 48 .