求解答两道prep，做了很久。。

zflucy11 · 发表于 2010-12-15 22:45:09

【1】A certain city with a population of 132,000 is to be divided into 11 voting districts, and no district is to have a population that is more than 10 percent greater than the population of any other district. What is the minimum possible population that the least populated district could have?
A.10700
B.10800
C.10900
D.11000
E.11100

【2】For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100)+1, then p is
Between 2 to 10
Between 10 to 20
Between 20 to 30
Between 30 to 40
Greater than 40

好久没做过数学题脑袋生锈了。。求助CDer，祝版友都拿到想要的成绩和offer!

sdcar2010 · 发表于 2010-12-15 23:21:43

A certain city with a population of 132,000 is to be divided into 11 voting districts, and no district is to have a population that is more than 10 percent greater than the population of any other district. What is the minimum possible population that the least populated district could have?
A.10700
B.10800
C.10900
D.11000
E.11100

The worst scenario is that 1 district has the lowest population of n while any of the other 10 districts has 1.1n. In this way, the requirement that no district is to have a population that is more than 10 percent greater than the population of any other district is fulfilled.

n + 11 n = 132000; n = 132000/12 = 11000

D is the answer.

sdcar2010 · 发表于 2010-12-15 23:28:10

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100)+1, then p is
Between 2 to 10
Between 10 to 20
Between 20 to 30
Between 30 to 40
Greater than 40

h(100) = 2*4*6*8*. . .*94*96*98*100
=(2^50)*1*2*3*4*. . .*47*48*49*50

So prime factors of h(100) includes all the prime numbers between 2 and 47.

It is also known that any two consecutive natural number are co-prime, meaning that these two numbers do not share any factors. Therefore h(100) and h(100)+1 are co-prime, so they do not share any prime number as their co-factors. Therefore, h(100) + 1 does not contain any prime factor between 2 and 47. Thus, the smallest prime factor for h(100) + 1 greater than 47.

zflucy11 · 发表于 2010-12-15 23:47:14

@@看得目瞪口呆……思路好清晰啊，谢谢大牛！
相邻的两个整数没有相同的质因子这条好像真的从来没学过。。

唱什么歌 · 发表于 2011-4-30 22:16:57

受教~

wxch · 发表于 2011-5-1 10:05:54

受益匪浅谢谢啦

charles1987 · 发表于 2011-5-1 15:57:13

但是我怎么算出来第一题是 B.
looking for OA .

wangjieava23 · 发表于 2011-5-1 16:34:08

领教了，什么叫大牛。

WO爱逍遥 · 发表于 2011-5-1 17:19:44

爆牛啊

求解答两道prep，做了很久。。

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