Number properties ?

[sapuna]

For every positive integer n, the function h(n) is defined as 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 :

1)between 2 and 10
2)between 10 and 20
3)between 20 and 30
4)between 30 and 40
5)greater than 40

[Brent@GMATPrepNow]

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, the p is

A: Between 2 & 10
B: Between 10 & 20
C: Between 20 & 30
D: Between 30 & 40
E: Greater than 40

Important Concept: If integer k is greater than 1, and k is a factor (divisor) of N, then k is not a divisor of N+1
For example, since 7 is a factor of 350, we know that 7 is not a factor of (350+1)
Similarly, since 8 is a factor of 312, we know that 8 is not a factor of 313

Now let's examine h(100)
h(100) = (2)(4)(6)(8)....(96)(98)(100)
= (2x1)(2x2)(2x3)(2x4)....(2x48)(2x49)(2x50)
Factor out all of the 2's to get: h(100) = [2^50][(1)(2)(3)(4)....(48)(49)(50)]

Since 2 is in the product of h(100), we know that 2 is a factor of h(100), which means that 2 is not a factor of h(100)+1 (based on the above rule)

Similarly, since 3 is in the product of h(100), we know that 3 is a factor of h(100), which means that 3 is not a factor of h(100)+1 (based on the above rule)

Similarly, since 5 is in the product of h(100), we know that 5 is a factor of h(100), which means that 5 is not a factor of h(100)+1 (based on the above rule)

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Similarly, since 47 is in the product of h(100), we know that 47 is a factor of h(100), which means that 47 is not a factor of h(100)+1 (based on the above rule)

So, we can see that none of the primes from 2 to 47 can be factors of h(100)+1, which means the smallest prime factor of h(100)+1 must be greater than 47.

Answer = E

Cheers,
Brent

[sapuna]

Thank you dude, you`re such a machine. I don`t know whether you`ve come across this exact problem in the past but you sure answered me quickly ! ;]

1 more thing - shouldn`t we say that the smallest prime factor of h(100)+ 1 should be greater than 50 , since 50 is in the product of h(100) and therefore it won`t be a factor of h(100+1) ?

And if the question was, which exactly is the smallest prime number, thats factor, the answer would be 53 ?

Cheers

[Brent@GMATPrepNow]

That may be the most common question on this forum, and most of the experts here have answered it several times (in fact, expect a few more posts from them with their take on the problem ).

You're right; the smallest prime factor will be greater than 50, but that doesn't mean that E is incorrect. Also, keep in mind that an answer choice that reads greater than 50 might be a tip-off to guessers since 50 is half of 100.

Regarding your question of whether or not the correct answer is 53, I'm not sure.
(2)(4)(6)(8)....(96)(98)(100) + 1 is a HUGE number, and don't see a quick way to determine whether or not it will be divisible by 53. That said, I'm going to guess that it IS divisible by 53, but I could be wrong.

Cheers,
Brent

[GMATinsight]

For every positive integer n, the function h(n) is defined as 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 :

1)between 2 and 10
2)between 10 and 20
3)between 20 and 30
4)between 30 and 40
5)greater than 40

As per the definition of the question

h(100) = 2 x 4 x 6 x 8 x 10 x 12 x 14 ... and so on...98 x 100 (Total 50 terms)

=> h(100) = 2^50 (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10......and so on...x 48 x 49 x 50)

This means h(100) is multiple of all prime numbers between 1 and 50

therefore h(100)+1 will leave a remainder of 1 when divided by any prime number from 1 to 50

therefore, p, which is a factor of h(100)+1, will certainly be greater than a prime numbers greater than 50

Hence, "p" must be greater than 40 as per the following options

Answer: Option - E

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Hi sapuna,

This question shows up every so often in this Forum and it is definitely tougher than a typical GMAT question.

As a general rule, Quant questions are almost always based on a pattern of some kind (math formula, math rule, Number Property, etc.). If you can't immediately deduce a pattern, then you might have to "play around" a bit with the question to try to deduce what the pattern is. In the broad sense, it's critical thinking: here's a weird situation - what can I do to figure it out?

Based on the description of the function in the prompt, we can run some "TESTS" to try to figure things out....

The H(n) is the product of all the even integers from 2 to n, inclusive.

So....
H(4) = 2x4 = 8
The prime factors of 8 are (2)(2)(2)
If we do H(4) + 1 = 9, then the prime factors are (3)(3)
NOTICE how NONE of the prime factors of 8 are in 9? That's interesting....

H(6) = 2x4x6 = 48
The prime factors of 48 are (2)(2)(2)(2)(3)
If we do H(6) + 1 = 49, then the prime factors are (7)(7)
NOTICE how NONE of the prime factors of 48 are in 49? That's interesting....and probably a pattern, since it's happened TWICE NOW.

From here, I'd have to deduce that this pattern holds true. With H(100), I know that there are LOTS of primes that go in (the largest of which is 47, which can be "found" in 94). I have to assume that NONE of them will go into H(100) + 1. Thus, the smallest prime would have to be greater than 47. The question doesn't actually ask us for the exact answer though (the answer choices are "ranges").

The takeaway from all of this is that you shouldn't be afraid to "play" with a question a bit. In the end, you don't have to be a brilliant mathematician to answer this question, but you're also not allowed to just stare at it either.

GMAT assassins aren't born, they're made,
Rich