HMMT 十一月 2013 · THM 赛 · 第 9 题

英文解析

9. [ 7 ] Let N be the largest positive integer that can be expressed as a 2013-digit base − 4 number. Whatis the remainder when N is divided by 210?
   ∑
   1007 1007
   1006
   16 − 1 16 − 1
   2 i
   Answer: 51 The largest is 3 · 4 = 3 = .
   i =0
   16 − 1 5
   1007 8 2
   This is 1 (mod 2), 0 (mod 3), 3 · 1007 ≡ 21 ≡ 1 (mod 5), and 3(2 − 1) ≡ 3(2 − 1) ≡ 3(2 − 1) ≡ 2
   (mod 7), so we need 1 (mod 10) and 9 (mod 21), which is 9 + 2 · 21 = 51 (mod 210).