HMMT 二月 2018 · ALGNT 赛 · 第 3 题

HMMT February 2018 — ALGNT Round — Problem 3

There are two prime numbers p so that 5 p can be expressed in the form for some positive integer 5 n . What is the sum of these two prime numbers?

解析

There are two prime numbers p so that 5 p can be expressed in the form for some positive integer 5 n . What is the sum of these two prime numbers? Proposed by: Kevin Sun Answer: 52 2 2 Note that the remainder when n is divided by 5 must be 0, 1, or 4. Then we have that 25 p = n 2 2 or 25 p = n − 1 or 25 p = n − 4. In the first case there are no solutions. In the second case, if 25 p = ( n − 1)( n + 1), then we must have n − 1 = 25 or n + 1 = 25 as n − 1 and n + 1 cannot both be divisible by 5, and also cannot both have a factor besides 25. Similarly, in the third case, 25 p = ( n − 2)( n + 2), so we must have n − 2 = 25 or n + 2 = 25. Therefore the n we have to check are 23 , 24 , 26 , 27. These give values of p = 21, p = 23, p = 27, and p = 29, of which only 23 and 29 are prime, so the answer is 23 + 29 = 52.