# Thread: Function and prime factors

1. ## Function and prime factors

Hi! In studying for the GMAT, I ran into a rather difficult problem. I don't know where to start or any of the steps to choosing the correct 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 smallest prime factor of h(100)+1, then p is:

Between 2 and 10
Between 10 and 20
Between 20 and 30
Between 30 and 40
Greater than 40

The correct answer is Greater than 40, but I am at a loss as to how to find it. Any help would be greatly appreciated.

2. ## Re: Function and prime factors

Note that $\displaystyle h(100) = 2^{50}\cdot50!$. The proof that h(100) + 1 is not divisible by any number <= 40 is similar to the Euclid's proof that there are infinitely many primes.

3. ## Re: Function and prime factors

Originally Posted by emakarov
Note that $\displaystyle h(100) = 2^{50}\cdot50!$. The proof that h(100) + 1 is not divisible by any number <= 40 is similar to the Euclid's proof that there are infinitely many primes.
Ah I see, thank you very much for your explanation.