HMMT 二月 2016 · 冲刺赛 · 第 17 题
HMMT February 2016 — Guts Round — Problem 17
题目详情
- [ 11 ] Compute the sum of all integers 1 ≤ a ≤ 10 with the following property: there exist integers p 2 2 and q such that p , q , p + a and q + a are all distinct prime numbers.
解析
- [ 11 ] Compute the sum of all integers 1 ≤ a ≤ 10 with the following property: there exist integers p 2 2 and q such that p , q , p + a and q + a are all distinct prime numbers. Proposed by: Evan Chen Answer: 20 Odd a fail for parity reasons and a ≡ 2 (mod 3) fail for mod 3 reasons. This leaves a ∈ { 4 , 6 , 10 } . It is easy to construct p and q for each of these, take ( p, q ) = (3 , 5) , (5 , 11) , (3 , 7), respectively.