prep裏面有道數學題不會。。。請高人指點！~

cherryser

For every positive enven 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

a. between 2 and 10
b. between 10 and 20
c. between 20 and 30
d. between 30 and 40
e. greater than 40

正確答案是E。。。但是爲什麽？高人教教我吧~~！

stevenckguo

选A, 对不对?

如果我选对了,我再告诉你,我的思路.

答案是什么

fatmoe

google出来的解法
Guest, this is definitely a difficult number properties question. Let's first consider the prime factors of h(100). According to the given function,
h(100) = 2*4*6*8*...*100
By factoring a 2 from each term of our function, h(100) can be rewritten as
2^50*(1*2*3*...*50).
Thus, all integers up to 50 - including all prime numbers up to 50 - are factors of h(100).
Therefore, h(100) + 1 cannot have any prime factors 50 or below, since dividing this value by any of these prime numbers will yield a remainder of 1.
Since the smallest prime number that can be a factor of h(100) + 1 has to be greater than 50, The correct answer is E.

zxc1118zxc

prep数学有解析啊 那个繁体版的

假設h(100)+1最小因數是 n , n<50
例如47好了, 既然47是因數, 他就能整除h(100)+1
可是發現h(100)裡面有47的因數,可以整除 那h(100)+1被47除的餘數為1,表示47不是h(100)+1的因數
同理去想 小於50的其它因式 也是會一樣的結果
所以h(100)+1的因式是在 大於50外
重點在那50!, 假如 小於50的質數是h(100)的factor,那就不會是h(100)+1的factor ,因為餘數會是1

cherryser

I SEE, THANKS A LOT @