请教道PREP的数学题

For every positive even integer, 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, the P is

1 between 2 and 10
2 between 10 and 20
3 between 20 and 30
4 between 30 and 40
5 greaster than 40

答案是第五个……

麻烦问下这道题具体应该怎么推，然后得出答案。。

这道题目这么考虑，对于任意小于40的质数来说，它的两倍必然是一个小于80的偶数，这个偶数必然可以被h(100)整除，那么这个质数也可以被h(100)整除，加上1之后余数就是1，不能整除了，所以p必然是大于40的，我觉得更强的条件应该是大于50的