PUMaC 2012 · 数论（A 组） · 第 3 题

PUMaC 2012 — Number Theory (Division A) — Problem 3

[ 4 ] Let the sequence { x } be defined by x ∈ { 5 , 7 } and, for k ≥ 1 , x ∈ { 5 , 7 } . For n 1 k +1 5 7 5 7 5 7 5 7 5 5 7 7 5 5 7 7 example, the possible values of x are 5 , 5 , 5 , 5 , 7 , 7 , 7 , and 7 . Determine the 3 sum of all possible values for the last two digits of x . 2012

解析

[ 4 ] Let the sequence { x } be defined by x ∈ { 5 , 7 } and, for k ≥ 1 , x ∈ { 5 , 7 } . For n 1 k +1 5 7 5 7 5 7 5 7 5 5 7 7 5 5 7 7 example, the possible values of x are 5 , 5 , 5 , 5 , 7 , 7 , 7 , and 7 . Determine the 3 sum of all possible values for the last two digits of x . 2012 4 n Solution: Note that 7 = 2401 ≡ 1 (mod 100) and that 5 ≡ 25(mod 100) for n ≥ 2. Then we must consider 3 cases. 1 x Case 1: We consider numbers of the form 5 , where x is an odd positive integer greater than x