Smallest prime factor question...?

Question

I was looking at some GMAT sample questions when I ran across this one and hoped someone could explain how to arrive at the answer...

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 smalled 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

The answer is E.

Sorry, the end of the question should read: If p is the smallest prime factor of h(100)+1, then p is

Answer 1

Think of h(100). It is 2 x 4 x 6 x ... x 98 x 100.

This is the same as 2 (1 x 2 x 3 x ... x 49 x 50) or just 2 x 50!

Because you have all the numbers from 1 to 50 in the product:
If you divide it by 2, you get 0 as a remainder
If you divide it by 3, you get 0 as a remainder
If you divide it by 4, you get 0 as a remainder
etc.

Now add 1 to this whole thing so you have h(100) + 1

If you divide it by 2, you are going to get a remainder of 1.
If you divide it by 3, you are going to get a remainder of 1.
If you divide it by 4, you get a remainder of 1
etc.

All the way to at least 50...

For an integer to be a factor, it must divide evenly without a remainder. We have no integer factors before 50, so 'p', the smallest prime factor of h(100) + 1, will definitely be greater than 40.

The answer is E, as you said.

Answer 2

You will feel stupid when I show you. h(100)=2*4*6*8...100
h(100)/(2*2*2...)= 1*2*3*4...50

let x=h(100)

if x is divisable by 2 then p is not because x+1=p is the next number every other number is divisable by 2.
if x is divisable by 3 p is not because it is the next (not the third next number)
if x is divisable by 5 p is not it is not five greater(only one more)
if x is divisable by 7 p..................
look above x is divisable by 1-50. the answer is f. greater than 50