Fei wrote:Hi

I just took the GMAT prep and found that there was little overlap. I got 720. I wish the solutions were provided though because I came across a few hairy questions.

Q3 ( for the third question, I don't consider this easy, and I am a bit worried about running into this on the real thing so early on).

**h(n) is the product of even integers from 2 to n, if p is the smallest prime factor of h(100) +1, then p is**

A. between 2&10, B between 10&20, C between 20&30, D between 30&40, E greater than 40.

I appreciate anyone's thought on the above questions, thanks beatthegmat.

Here's the number property rule that's being tested with this problem:

**If x is a positive integer, the only factor common both to x and to x+1 is 1; they share no other factors. Any factor of x (other than 1) will NOT be a factor of x+1.**
Let's examine why:

If x is a multiple of 2, what's the next largest multiple of 2? If x=4, the next largest multiple of 2 is 4+2=6.

**So the next largest multiple of 2 is x+2.**
If x is a multiple of 3, what's the next largest multiple of 3? If x=6, the next largest multiple of 3 is 6+3=9.

**So the next largest multiple of 3 is x+3.**
Using this logic, if we go from x to x+1, we get only to the next largest multiple of 1.

**So 1 is the only factor common both to x and to x+1.**
Thus, in the problem above, we know that 1 is the only factor common both to h(100) and to h(100) + 1.

*They share no other factors*.

h(100) = 2 * 4 * 6 *....* 94 * 96 * 98 * 100

Factoring out 2, we get:

h(100) = 2^50 (1 * 2 * 3 *... * 47 * 48 * 49 * 50)

Looking at the set of parentheses on the right, we can see that every prime number between 1 and 50 is a factor of h(100). This means that

*none* of the prime numbers between 1 and 50 is a factor of h(100) + 1, because

**h(100) and h(100) + 1 share no factors other than 1**.

So the smallest prime factor of h(100) + 1 must be

*greater than 50*.

**The correct answer is E.**
This is a hard and strange question. Most test-takers would be wise to guess and move on to more familiar questions. The good news is that if you get all of the other questions right, skipping this question will have little impact on your score. And remember: a quarter of the questions are experimental and do not affect your score at all.