HMMT 二月 2010 · 冲刺赛 · 第 7 题
HMMT February 2010 — Guts Round — Problem 7
题目详情
英文原题
- [ 6 ] What are the last 8 digits of
11 × 101 × 1001 × 10001 × 100001 × 1000001 × 111?
解析
英文解析
- [ 6 ] What are the last 8 digits of
11 × 101 × 1001 × 10001 × 100001 × 1000001 × 111?
Answer: 19754321 Multiply terms in a clever order.
11 · 101 · 10001 = 11 , 111 , 111 ,
111 · 1001 · 1000001 = 111 , 111 , 111 , 111 .
The last eight digits of 11 , 111 , 111 · 111 , 111 , 111 , 111 are 87654321. We then just need to compute thelast 8 digits of 87654321 · 100001 = 87654321 + . . . 32100000, which are 19754321.