GMAT, GRE Factors, Primes, Functions and [ Number Properties ] Problem

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?

A. Between 2 and 20
B. Between 10 and 20
C. Between 20 and 30
D. Between 30 and 40
E. Greater than 40 



h(100)+1=2∗4∗6∗...∗100+1=250∗(1∗2∗3∗..∗50)+1=250∗50!+1$h(100)+1=2\ast 4\ast 6\ast ...\ast 100+1=2^{50}\ast (1\ast 2\ast 3\ast ..\ast 50)+1=2^{50}\ast 50!+1$

We have two numbers 

h(100)=250∗50!$h(100)=2^{50}\ast 50!$ and h(100)+1=250∗50!+1$h(100)+1=2^{50}\ast 50!+1$ are consecutive integers. 

Two consecutive integers are co-prime, and co-primes don't share any common factor except 1. The same holds true for integers. Both 50 and 51 are consecutive integers, thus only common factor they share is 1.

h(100)=250∗50!$h(100)=2^{50}\ast 50!$ has all prime numbers from 1 to 50 as its factors, according to above h(100)+1=250∗50!+1$h(100)+1=2^{50}\ast 50!+1$ will not have ANY prime factor from 1 to 50. Hence p$p$ (>1$>1$), the smallest prime factor of h(100)+1$h(100)+1$ will be more than 50.

E