Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

28 Jan 2012, 17:46

32

This post received KUDOS

257

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

65% (hard)

Question Stats:

62% (00:45) correct
38% (01:15) wrong based on 1304 sessions

HideShow timer Statistics

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?

A. Between 2 and 20 B. Between 10 and 20 C. Between 20 and 30 D. Between 30 and 40 E. Greater than 40

This is how I am trying to solve this. Please help me if you think I am not right. OA is not provided in the book.

h(100) = 2 * 4 * 6 ****************100

Tn = a1 + (n-1) d-----------------------(1) where Tn is the last term, a1 is the first term and d is the common difference of the evenly spaced set.

100 = 2 + (n-1) 2 n = 50

Product of terms = Average * number of terms

Average = (a1+an)/2 Therefore average = 102/2 = 51 Product of the series = 51*50 = 2550.

H(100) + 1 = 2550+1 = 2551 which is prime. And prime numbers have exactly 2 factors 1 and the number itself. Therefore for me D is the answer i.e. < 10

h(n) is the product of the even numbers from 2 to n, inclusive, and p is the least prime factor of h(100)+1. What is the range of p?

< 40 < 30 > 40 < 10 Indeterminate

Below is the proper version of this 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? A. between 2 and 20 B. between 10 and 20 C. between 20 and 30 D. between 30 and 40 E. greater than 40

Now, two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

As \(h(100)=2^{50}*50!\) has all prime numbers from 1 to 50 as its factors, according to above \(h(100)+1=2^{50}*50!+1\) won't have ANY prime factor from 1 to 50. Hence \(p\) (\(>1\)), the smallest prime factor of \(h(100)+1\) will be more than 50.

Bunuel - how did you get h(100) = 2^50 * 50! ?? Sorry for been a pain.

Given that the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive, so: \(h(100)=2*4*6*...*100=(2*1)*(2*2)*(2*3)*(2*4)*...*(2*50)\) --> factor out all 50 2's: \(h(100)=2^{50}*(1*2*3*..*50)=2^{50}*50!\).

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

02 Mar 2012, 13:26

1

This post received KUDOS

Thanks Bunuel. It was very helpful. This was my 3rd question on the practice test, I spent a minute without even knowing where to start from, so I made a random guess and moved on. I got 13 incorrect questions out of the 37, but I managed to score 48. thanks again

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

01 May 2012, 06:42

1

This post received KUDOS

Thanks for the explanation. I also got this as my 3rd question, couldn't figure it out, ended up guessing and moving on. Surprising that GMATPrep throws you such hard questions early.

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

10 May 2012, 06:53

1

This post received KUDOS

1

This post was BOOKMARKED

Bunuel wrote:

enigma123 wrote:

h(n) is the product of the even numbers from 2 to n, inclusive, and p is the least prime factor of h(100)+1. What is the range of p?

< 40 < 30 > 40 < 10 Indeterminate

Below is the proper version of this 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? A. between 2 and 20 B. between 10 and 20 C. between 20 and 30 D. between 30 and 40 E. greater than 40

Now, two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

As \(h(100)=2^{50}*50!\) has all prime numbers from 1 to 50 as its factors, according to above \(h(100)+1=2^{50}*50!+1\) won't have ANY prime factor from 1 to 50. Hence \(p\) (\(>1\)), the smallest prime factor of \(h(100)+1\) will be more than 50.

Answer: E.

That is an impressive solution. <-- understatement

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?

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

This is how I am trying to solve this. Please help me if you think I am not right. OA is not provided in the book.

h(100) = 2 * 4 * 6 ****************100

Tn = a1 + (n-1) d-----------------------(1) where Tn is the last term, a1 is the first term and d is the common difference of the evenly spaced set.

100 = 2 + (n-1) 2 n = 50

Product of terms = Average * number of terms

Average = (a1+an)/2 Therefore average = 102/2 = 51 Product of the series = 51*50 = 2550.

H(100) + 1 = 2550+1 = 2551 which is prime. And prime numbers have exactly 2 factors 1 and the number itself. Therefore for me D is the answer i.e. < 10

Karishma, could you recommend me some other on-line material or sites where I can study these kind of problems in-depth? Thank you!

Check out part i of the link given above as well. That should cover the theory that is useful for such questions. As for examples, you can search for 'factors consecutive integers' in the Quant forum and you should hit quite a few questions based on these concepts. I got one in one of my posts: if-n-is-a-positive-integer-and-r-is-the-remainder-when-119518.html?hilit=factor%20consecutive%20integers _________________

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

29 May 2012, 20:30

hifunda wrote:

Thanks for the explanation. I also got this as my 3rd question, couldn't figure it out, ended up guessing and moving on. Surprising that GMATPrep throws you such hard questions early.

this was my second question on the first practice test, i'll have to review this answer quite a few times until it sticks. Great answer

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

27 Aug 2012, 05:07

1

This post received KUDOS

Hey Bunuel, I used the following logic, and got E. (2*4*6*8.... 100) +1 is of the format 6p +1. Therefore, its a prime number and has only two factors, one and itself. So, its definitely greater than 40.

Hey Bunuel, I used the following logic, and got E. (2*4*6*8.... 100) +1 is of the format 6p +1. Therefore, its a prime number and has only two factors, one and itself. So, its definitely greater than 40.

So, E

Is it correct?

Every prime number greater than 3 is of one of the two forms: (6a -1), (6a + 1) e.g. 5 = 6*1 - 1; 7 = 6*1 + 1; 11 = 6*2 - 1; 13 = 6*2 + 1 etc

But every number of the form (6a -1) or (6a + 1) is not prime. e.g. 25 = 6*4 + 1 i.e. it is of the form 6a+1. But 25 is not prime.

So just because a number is of the form 6a + 1, we cannot say that it is definitely prime.
_________________

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

02 Sep 2012, 11:31

4

This post received KUDOS

1

This post was BOOKMARKED

I'm not taking credit for the following solution. Found it in another forum.

Two consecutive integers cannot be divisible by the same integer greater than 1.

So if we can prove that h(100) is divisible by every number smaller than 50, we proved that h(100)+1 is NOT divisible by any number smaller than 50 (besides 1).

2*4*6*...*98*100=2*(1*2*3*...*50)=2*(50!) Hence, h(100) is divisible by every number smaller than 51. So the smallest prime factor of h(100)+1 is at least 53.

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

21 Sep 2012, 01:49

3

This post received KUDOS

Got this one on the 9th question of my GMATPrep test. I actually reached a good point on this problem.

I got to the point that the product of 2*4*6*....*100 = (2^50)*50!

At this point I lost the plot and developed a weird approach here.

Since the last number in the given h(n) was 100 so the actual value could be like xxx00. So now, h(n) + 1 would be some value like xxx01.

So the possible prime factors are effectively 3, 7, 11, 13 etc.

I simply gambled on one of these till 20 being a factor and chose A. It was wrong though. The correct answer is indeed E.

The explanations above specially the one by Bunuel where the point of co-prime numbers not having any common factors but 1 was particularly good stuff and I had not read it before. This information will surely help the next time provided I am able to fully recognize it down.
_________________

My attempt to capture my B-School Journey in a Blog : tranquilnomadgmat.blogspot.com

Re: For every positive even integer n, the function h(n) is defined to be [#permalink]

Show Tags

22 Apr 2013, 22:40

2

This post received KUDOS

2

This post was BOOKMARKED

h(n) = the product of all even integers from 2 to n inclusive => h(100) + 1 = (2 * 4 * 6 * 8 * ....... * 100) + 1 = 2^50 * (1*2*3*4*......*50) + 1

As all the numbers from 2 to 50 are factors of 2^50 * (1*2*3.....*50), none of these can be factors of 2^50 * (1*2*3*4*......*50) + 1 (i.e. of h(100)+1)

Therefore the smallest prime factor of h(100) + 1 is greater than 50.

For every positive 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

A) between 2 and 10 B) between 10 and 20 C) between 20 and 30 D) between 30 and 40 E) Greater than 40

Consider the second part of the question. We need to find a bound for the smallest prime factor of h(100)+1. Any number p that is a factor of h(100)+1, leaves a remainder of (p-1) when it divides h(100).

remainder(h(100)/p) = p-1.

Another way of saying this is "p is a prime number that is not a divisor of h(100)" (B)

From (A):

Clearly h(100) is divisible by all prime numbers less than 50 (as h(100) is divisible by 50!). The smallest prime that is not a divisor of h(100) should therefore be greater than 50.

Looking at the choices, the most appropriate choice is:

(e) greater than 40.

Hope that Helps!
_________________

Karthik P. Founder and CEO, August Academy.

We are an India based MBA admissions consulting firm. We've been helping Indian MBA applicants (students belonging to the most competitive MBA admissions pool) since 2013. Several of our students study at top 20 schools in the US including Kellogg, MIT, Yale, Ross and Darden, top European schools including London Business School and Oxford, top Canadian schools like Rotman and top Asian schools like ISB and NUS. Our students have won over USD 2.1 Million in scholarships over the last 3 admissions years.

Last term number n=25 Average = 26 Sum = 26*25=650

h(50)+1 = 651 which is not a prime number as it is divisible by 3. Please correct me if i m wrong here. my answer is A.

I am not sure where you got this question since the original question here talks about h(100) + 1. Even if we do assume h(50) + 1, note that h(2) = 2 h(4) = 2*4 h(6) = 2*4*6 .... So I am not sure how you got average 26. The correct answer here is (E) as explained in the link given in my post above.
_________________