PUMaC 2014 · 数论（A 组） · 第 6 题

PUMaC 2014 — Number Theory (Division A) — Problem 6

[ 6 ] Given S = { 2 , 5 , 8 , 11 , 14 , 17 , 20 ... } . Given that one can choose n different numbers from S , ∑ 1 n { A , A , ...A } , s.t. = 1. Find the minimum possible value of n. 1 2 n i =1 A i

解析

[ 6 ] Given S = { 2 , 5 , 8 , 11 , 14 , 17 , 20 ... } . Given that one can choose n different numbers from S , ∑ 1 n { A , A , ...A } , s.t. = 1. Find the minimum possible value of n. 1 2 n i =1 A i Solution: ∑ ∏ 1 n n It is clearly not possible that n ≤ 5. We see that = 1 can be rewritten as A = i i =1 i =1 A i ∑ ∏ n n n − 1 A . Taking mod 3, we have 2 ≡ n 2 which reduces to n ≡ 2 mod 3. Hence the j i =1 j 6 = i smallest possible n = 8 and such an example for 8 will be: { 2 , 5 , 8 , 11 , 20 , 44 , 89 , 792 } .