taneja.niks wrote:For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n inclusive. If p is the smallest prime factor of h(100) + 1, then p is between
1) 2 and 10
2) 10 and 20
3) 20 and 30
4) 30 and 40
5) > 40


Answer >40

how to proceed such questions trying to divide will be very long and time consuming advice a smaller way for it

Here is the rule that is 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.

Let's examine why:

If x is a multiple of 2, the next largest multiple of 2 is x+2.
If x is a multiple of 3, 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. They share no other factors. (For future reference, integers that share no factors other than 1 are called coprimes.)

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.

GMATGuruNY wrote: 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.

hi guru,
as usual excellent solution.
well i think this question was asked before some days as well.
i am bit confused.
The answer is >50
and in option we have no greater than 50
we have greater than 40 as well
i agree >50 is >40 as well
but
if we take 42,43,44 for example,it will be correct from answers only(as answer is E >40) but process wise it wont be correct.
So are the options of this question not well formed?
Just curious.

thanks

frank1 wrote:

GMATGuruNY wrote: 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.

hi guru,
as usual excellent solution.
well i think this question was asked before some days as well.
i am bit confused.
The answer is >50
and in option we have no greater than 50
we have greater than 40 as well
i agree >50 is >40 as well
but
if we take 42,43,44 for example,it will be correct from answers only(as answer is E >40) but process wise it wont be correct.
So are the options of this question not well formed?
Just curious.

thanks

option is wrong p cant be any prime number between 1 to 50 So, it should be p > 50

sixpointer wrote:
option is wrong p cant be any prime number between 1 to 50 So, it should be p > 50

I remember it was same even last time it was asked.(in other posts)
so i thought question is same in the source material.
Is it?
topic poster may have better idea about it.