PUMaC 2012 · 团队赛 · 第 8 题

PUMaC 2012 — Team Round — Problem 8

(3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred: • Mom: Tell me the number of zeros at the end of 2012! • PT: Huh? 2012 ends in 2, so there aren’t any zeros. • Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about 2012, I was asking about 2012!. What is the correct answer?

解析

Problem: (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred: • Mom: Tell me the number of zeros at the end of 2012! 5 • PT: Huh? 2012 ends in 2, so there aren’t any zeros. • Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about 2012, I was asking about 2012!. What is the correct answer? Answer: 501 Solution: We want to find the highest power of 10 which divides 2012! = 2012 · 2011 · n n n 2010 · · · 2 · 1. If 10 | 2012!, then 2 | 2012! and 5 | 2012!. 2012! clearly contains more powers n of 2 than powers of 5, so we want to find the largest integer n such that 5 | 2012!. Writing 2012! = 2012 · 2011 · 2010 · · · 2 · 1, we can take a factor of 5 from each of 5 , 10 , 15 , . . . , 2010. Then we can take another factor of 5 from 25 , 50 , 75 , . . . , 2000. We continue this process, to 2012 2012 2012 2012 give us n = b c + b c + b c + b c = 501 5 25 125 625 Author: Alan