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Prime factor problem I had trouble with on a practice exam

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Prime factor problem I had trouble with on a practice exam

by Sent » Sat Mar 26, 2016 8:09 am
Hi all, I recently took one of the official GMAT practice tests. There was one problem where I am still not sure how to get the answer. I was hoping someone could help explain!

Question:
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.

I actually guessed correctly on this problem but I still have no idea how to legitimately arrive at the answer

Any help is appreciated.
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by DavidG@VeritasPrep » Sat Mar 26, 2016 8:16 am
Sent wrote:Hi all, I recently took one of the official GMAT practice tests. There was one problem where I am still not sure how to get the answer. I was hoping someone could help explain!

Question:
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.

I actually guessed correctly on this problem but I still have no idea how to legitimately arrive at the answer

Any help is appreciated.
I wrote a blog post about a very similar question here: https://www.veritasprep.com/blog/2016/0 ... -the-gmat/
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by Brent@GMATPrepNow » Sat Mar 26, 2016 9:25 am
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)

.
.
.
.
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

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Brent
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by [email protected] » Sat Mar 26, 2016 9:30 am
Hi Sent,

This question tends to trouble most Test Takers 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 prime number though - it just asked for the range that the prime would fall into.

Final Answer: E

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
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by Sent » Sun Mar 27, 2016 7:57 am
Hi all, thanks for your responses! I like David & Brent's solutions for being the most intuitive (and Brent's "important concept", which is obvious when you think about it, but not as obvious if you have to come up with it yourself to solve a problem like this). I think Rich's way of running tests is useful too, and want to try to do this more. I often feel like I don't have the time to do this in the GMAT proper - which often leads to me getting some pattern recognition-type problems wrong!
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by [email protected] » Sun Mar 27, 2016 11:04 am
Hi Sent,

75 minutes is enough time to reasonably attempt all of the questions in the section (even if you choose to guess on a question to move past it). Any pacing 'issue' that you might be facing does not exist on its own - it's the result of other issues. If you continuously reread prompts because you're not taking notes, then THAT would be a likely cause. Staring at the screen or your pad, while not actually doing any work, is also a likely cause.

1) Do you have to rush through questions to finish the Quant and/or Verbal sections? If you do, then how many do you tend to have to guess on?

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by Sent » Sun Mar 27, 2016 1:49 pm
Hi Rich, I do not have any pacing problems on the verbal section, but I do tend to run out of time on the quant section. On the same test where I got this problem from, I guessed on the last two problems of the quant section, but also got the two before those wrong as well because I was rushing. I think this is the largest issue for my quant section. I find that I can generally solve almost all problems if given the time, but for the harder problems some times I don't solve them in an efficient manner, leading to 4-5 minutes spent on the problem. I suppose there may be a little "staring at the screen", but I almost feel like that happens more to me in the verbal section for difficult sentence correction or critical reasoning problems.
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by Sent » Sun Mar 27, 2016 1:49 pm
Hi Rich, I do not have any pacing problems on the verbal section, but I do tend to run out of time on the quant section. On the same test where I got this problem from, I guessed on the last two problems of the quant section, but also got the two before those wrong as well because I was rushing. I think this is the largest issue for my quant section. I find that I can generally solve almost all problems if given the time, but for the harder problems some times I don't solve them in an efficient manner, leading to 4-5 minutes spent on the problem. I suppose there may be a little "staring at the screen", but I almost feel like that happens more to me in the verbal section for difficult sentence correction or critical reasoning problems.
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by Brent@GMATPrepNow » Sun Mar 27, 2016 2:07 pm
Sent wrote:Hi Rich, I do not have any pacing problems on the verbal section, but I do tend to run out of time on the quant section.
I suggest that you manage your time in batches of 5 questions. To that end, you can use the following Milestone Charts to keep you on track:
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These times are easily remembered so you can quickly jot them on your noteboard on test day.

This (and more) is covered in our free GMAT time management video at https://www.gmatprepnow.com/module/gener ... es?id=1244

Also, early in one's studies, it's a good idea to use a stopwatch to help get an idea of what 2 minutes feels like. This tip and others can be found in these articles on "making friends with time on the GMAT"
https://www.gmatprepnow.com/articles/mak ... %93-part-i
https://www.gmatprepnow.com/articles/mak ... 93-part-ii

Cheers,
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by [email protected] » Sun Mar 27, 2016 9:23 pm
Hi Sent,

If you're running into this pacing 'issue' often (spending 4-5 minutes on each of a handful of questions), then your score will likely be limited on the Quant section (and by extension, your Overall Score will also be limited). NO question is worth more than 3 minutes of your time in the Quant section, so if you're spending more than 3 minutes on a prompt, then you should take a really good look at WHY you spent so much time on it. It could be that you're taking the "long way" to answering the question, but it also could be that you stared at it far too long before deciding what to do. As far as your overall performance is concerned, it's better to 'dump' those questions than spend 3+ minutes on them (even if you get them correct, it hurts your overall score too much).

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