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 350,000 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:

Re: Properties of Numbers- Any short cuts? [#permalink]
18 Feb 2010, 06:27

2

This post received KUDOS

Expert's post

4

This post was BOOKMARKED

If n is a positive integer and n-squared is divisible by 72, then the largest positive integer that must divide n is A. 6 B. 12 C. 24 D. 36 E. 48

The largest positive integer that must divide \(n\), means for the least value of \(n\) which satisfies the given statement in the question. The lowest square of an integer, which is multiple of \(72\) is \(144\) --> \(n^2=144=12^2=72*2\) --> \(n_{min}=12\). Largest factor of \(12\) is \(12\).

OR:

Given: \(72k=n^2\), where \(k\) is an integer \(\geq1\) (as \(n\) is positive).

\(72k=n^2\) --> \(n=6\sqrt{2k}\), as \(n\) is an integer \(\sqrt{2k}\), also must be an integer. The lowest value of \(k\), for which \(\sqrt{2k}\) is an integer is when \(k=2\) --> \(\sqrt{2k}=\sqrt{4}=2\) --> \(n=6\sqrt{2k}=6*2=12\)

Problem Solving Question [#permalink]
10 May 2010, 19:33

I was hoping to get some clarification on Problem 169 from Quantitative Review 2nd Ed:

Q: If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is: A 6, B 12, C 24, D 36, E 48

n^2 is divisible by 72, but it must also be greater than 72. If n is an integer, then n^2 must be a perfect square. The factorization of 72 is (8)(9), so if it is multiplied by 2, it will be (2)(8)(9) = (16)(9) = 144, a perfect square. So n^2 must be at least 144 or a multiple of 144, which means that n must be 12 or a multiple of 12.

I know that Quantitative Review also has 12 as the answer, but I had a question: Since n must be 12 or a multiple of 12, why is it that 48 isn't a solution since its a multiple of 12 and 48 divides 48 and is also the greatest number amongst the solutions, especially because the question does not state 'largest integer other than n that divides n'? What is the concept that I am not getting?

Re: Problem Solving Question [#permalink]
18 May 2010, 07:47

ok,see in order to find he largest positive integer that must divide , means for lowest value of n^2 which is 144 , or N comes out to be 12, and now if you devide this by 48 then it would not come out to be an integer. Hence he largest integer must be 12.

Re: Problem Solving Question [#permalink]
22 May 2010, 01:35

1

This post received KUDOS

abc123def wrote:

I was hoping to get some clarification on Problem 169 from Quantitative Review 2nd Ed:

Q: If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is: A 6, B 12, C 24, D 36, E 48

n^2 is divisible by 72, but it must also be greater than 72. If n is an integer, then n^2 must be a perfect square. The factorization of 72 is (8)(9), so if it is multiplied by 2, it will be (2)(8)(9) = (16)(9) = 144, a perfect square. So n^2 must be at least 144 or a multiple of 144, which means that n must be 12 or a multiple of 12.

I know that Quantitative Review also has 12 as the answer, but I had a question: Since n must be 12 or a multiple of 12, why is it that 48 isn't a solution since its a multiple of 12 and 48 divides 48 and is also the greatest number amongst the solutions, especially because the question does not state 'largest integer other than n that divides n'? What is the concept that I am not getting?

Please help.

This question testing MUST or COULD

In this case n can be 12 or 36 or 48

But if take n =12 which is one of the condition (least possible value of n), and divide it by any integer greater than 12. the resulting number can never be a integer (question is asking the largest possible value that MUST divide n, in all cases). Hence it can only be 12

But if the question asks about the largest possible value which COULD divide n, in that case answer can be 48 (largest value in all answer). though it can be bigger than 48 also.

If n is a positive integer and n^2 is divisible by 72,then the largest positive integer that must divide n is:

A.6 B.12 C.24 D.36 E.48

\(72=2^3*3^2\) In order to find the largest integer that must divide n, since there is no upper bound on n, we should choose the smallest possible value of n. Given the prime factorisation of 72, it is easy to see, the smallest n^2 divisible by 72 would be \(n^2=2^4*3^2\), hence the smallest choice of n would be \(n=2^2*3=12\) Hence, for all possible n, the smallest value is 12 Hence, for all possible n, 12 always divides n, and is the largest such value to work for all n

Re: If n is a positive integer and n^2 is divisible by 72, then [#permalink]
20 Feb 2014, 13:50

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Re: If n is a positive integer and n^2 is divisible by 72, then [#permalink]
19 Apr 2014, 06:09

Given: 72k=n^2, where k is an integer >=1(as n is positive).

K cannot be = 1 since n is an integer and 72 is not perfect square. _________________

Rgds, TGC! _____________________________________________________________________ I Assisted You => KUDOS Please _____________________________________________________________________________

Re: If n is a positive integer and n^2 is divisible by 72, then [#permalink]
13 May 2015, 14:23

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Back to hometown after a short trip to New Delhi for my visa appointment. Whoever tells you that the toughest part gets over once you get an admit is...