请教prep一道数学题

nesstang

for every positive even interger n, the function h(n) is defined to be the product of all the even intergers from 2 to n. 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. over 40

LuckyStars

h(n)=2*4*6*8*....*100 +1
=2^50 *(1*2*3*...*50)+1

因为p is the smallest prime factor of h(100)+1，而2^50 *(1*2*3*...*50) 中的prime由于有后面的1*...*50已经大于50了
所以选E

nesstang

谢谢~

viennawong

没看太明白诶。。。。能说说1*...*50已经大于50了的具体意义是什么吗？感谢呀

bodkicker

我的思路是
h(n)=2*4*6*8*....*100 +1
如果前面除了加1的部分能被一个质数（prime factor）整除的话，那么加1一定不能被其整除。
前面那个式子的质因数为2,3,5.。。。47（h(n)=2*4*6*8*....*100 +1=（1*2）*（2*2）*（3*2）*（4*2）*....*（50*2）+1），所以符合条件的最小的那个prime factor一定大于47