Mind-bending problem

[kouranjelika]

Hey Guys,

So this set me back a couple of worthless minutes...any takers?

For every positive 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 10
B) between 10 and 20
C) between 20 and 30
D) between 30 and 40
E) greater than 40

I found a few explanations on the web, some stuff about double factorials (huh?) and all kinds of other sophisticated stuff which I believe to be outside the range of this test. Can someone come up with a good basic explanation?

Here's what I originally thought about:
So we are multiplying all these evens. We are going to get an even. Cool, so that number's smallest Prime will be 2. BUT wait, there is that 1. What happens when we add a 1 to that gigantic even number, it becomes odd. But how can we possibly know what SMALLEST prime will be. I tried to use smaller ranges and find some sort of pattern, as in:
n=2 2+1=3 smallest 3
n=4 2*4=8+1=9 smallest 3
n=6 2*4*6=48+1=49 smallest 7
n=8 2*4*6*8=384+1=385 smallest 5
n=10 10*8*6*4*2 = 3840 + 1 = 3841 smallest 23 (this one I just did now, I couldn't figure it out during the test)

In any case, sure there is some sort of increasing pattern, but from 6 to 8, the number actually goes down. Also this is such a lengthily process, I don't know how others are, but when I'm stressing, I'm not the fastest at calculations and this was def a stressful question. Plus once I hit 3841, I was dumbstruck.

I hope Ron you have some brilliant solution coming my way :)

Thanks!

[RonPurewal]

Please search the forum before posting; thanks.

These two threads currently have a total of 62 posts, so something in there probably answers your question.

for-every-positive-even-integer-n-the-function-h-n-t1152.html

post5477.html

If you still have questions, please post them on the existing thread(s).

Also, please title future posts per the forum rules (= first few words of the problem statement), to facilitate future searches. Thanks.