Problem Solving

Can someone help me with this question?
For every positive integer n, the function h(n) is defined to be the product of all 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) betwwen 30 and 40
e) greater than 40

I think the way to solve this is to find out the prime factors of h(100) which in this case are:

47, 43, 41, 37, ....., 2

How we know 47 is a factor?

47* 2 = 94 and since we know 94 is a factor so 47 is also a factor

Since all primes from 2 to 47 are factors of h(100) they can not be factors of h(100)+1 so the ans will be E

Let me know if this is correct.

Yes, the answer is E. I am not sure whether that's the right approach.