HMMT 十一月 2013 · THM 赛 · 第 9 题
HMMT November 2013 — THM Round — Problem 9
题目详情
- [ 7 ] Let N be the largest positive integer that can be expressed as a 2013-digit base − 4 number. What is the remainder when N is divided by 210?
解析
- [ 7 ] Let N be the largest positive integer that can be expressed as a 2013-digit base − 4 number. What is 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).