HMMT 二月 2016 · 代数 · 第 4 题

英文解析

4. Determine the remainder when
   ⌊ ⌋
   2015
   ∑ii =0252
   is divided by 100, where b x c denotes the largest integer not greater than x .
   Proposed by: Alexander Katz
   Answer: 14
   i φ (25) 20
   Let r denote the remainder when 2 is divided by 25. Note that because 2 ≡ 2 ≡ 1 (mod 25),
   r is periodic with length 20. In addition, we find that 20 is the order of 2 mod 25. Since 2 is never aiimultiple of 5, all possible integers from 1 to 24 are represented by r , r , ..., r with the exceptions of
   1 2 20
   ∑ ∑
   20 24
   5, 10 ,15, and 20. Hence, r = i − (5 + 10 + 15 + 20) = 250.
   i =1 i =1 i
   We also have
   ⌊ ⌋
   2015 2015
   i i
   ∑ ∑
   2 2 − ri
   25 25=
   i =0 i =0
   2015 2015
   ∑ ∑i
   2 ri = −
   25 25
   i =0 i =0
   1999 15
   2016
   ∑ ∑
   2 − 1 r ri i = − −
   25 25 25
   i =0 i =0
   ( )
   201615
   ∑
   2 − 1 250 ri = − 100 −
   25 25 25
   i =0
   201615
   ∑
   2 − 1 ri
   ≡ − (mod 100)
   25 25
   i =0
   ∑
   We can calculate r = 185, so 15
   i =0 i
   ⌊ ⌋
   2015
   i 2016
   ∑
   2 2 − 186
   ≡ (mod 100)
   25 25
   i =0
   φ (625) 500 2016 16 2016
   Now 2 ≡ 2 ≡ 1 (mod 625), so 2 ≡ 2 ≡ 536 (mod 625). Hence 2 − 186 ≡ 350
   2016 2016
   (mod 625), and 2 − 186 ≡ 2 (mod 4). This implies that 2 − 186 ≡ 350 (mod 2500), and so
   2016
   2 − 186
   ≡ 14 (mod 100).
   25