GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 20 Nov 2018, 05:48

Close

GMAT Club Daily Prep

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.

Close

Request Expert Reply

Confirm Cancel
Events & Promotions in November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
  • All GMAT Club Tests are Free and open on November 22nd in celebration of Thanksgiving Day!

     November 22, 2018

     November 22, 2018

     10:00 PM PST

     11:00 PM PST

    Mark your calendars - All GMAT Club Tests are free and open November 22nd to celebrate Thanksgiving Day! Access will be available from 0:01 AM to 11:59 PM, Pacific Time (USA)
  • Free lesson on number properties

     November 23, 2018

     November 23, 2018

     10:00 PM PST

     11:00 PM PST

    Practice the one most important Quant section - Integer properties, and rapidly improve your skills.

If n is a positive integer and n^2 is divisible by 72, then

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Intern
Intern
avatar
Joined: 27 Mar 2012
Posts: 6
If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 31 Mar 2012, 01:36
12
94
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

58% (00:52) correct 42% (01:07) wrong based on 1938 sessions

HideShow timer Statistics

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
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 31 Mar 2012, 01:47
26
1
49
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

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 \(k=2\) --> \(\sqrt{2k}=\sqrt{4}=2\) --> \(n=6\sqrt{2k}=6*2=12\)

Answer: B.

Similar problem:
division-factor-88388.html#p666722

Hope it's helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Most Helpful Community Reply
Manager
Manager
avatar
Joined: 28 Feb 2012
Posts: 112
Concentration: Strategy, International Business
Schools: INSEAD Jan '13
GPA: 3.9
WE: Marketing (Other)
GMAT ToolKit User
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 14 Aug 2012, 21:50
6
2
Normally i divide the number into the primes just to see how many more primes we need to satisfy the condition, so in our case:
n^2/72=n*n/2^3*3^2, in order to have minimum in denominator we should try modify the smallest number. If we have one more 2 then the n*n will perfectly be devisible to 2^4*3^2 from here we see that the largest number is 2*2*3=12
Hope i explained my thought.
_________________

If you found my post useful and/or interesting - you are welcome to give kudos!

General Discussion
Manager
Manager
avatar
Joined: 09 Jul 2010
Posts: 88
GMAT ToolKit User
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 21 Apr 2012, 13:04
Hi All

I have a confusion about this question

169. If n is a positive integer and n2 is divisible by 72, then
the largest positive integerthat must divide n is
(A) 6
(8) 12
(C) 24
(0) 36
(E) 48

If we are looking for largest positive integer that must divide n, why can it not be 48.

because if n2 = 72 * 32 then n will be 48 , so does this not mean n is divisible by 48.


Please explain.

Thanks
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 21 Apr 2012, 13:14
3
1
Merging similar topics.

raviram80 wrote:
Hi All

I have a confusion about this question

169. If n is a positive integer and n2 is divisible by 72, then the largest positive integer that must divide n is
(A) 6
(8) 12
(C) 24
(0) 36
(E) 48

If we are looking for largest positive integer that must divide n, why can it not be 48.

because if n2 = 72 * 32 then n will be 48 , so does this not mean n is divisible by 48.

Please explain.

Thanks


The question asks about "the largest positive integer that MUST divide n", not COULD divide n. Since the least value of n for which n^2 is a multiple of 72 is 12 then the largest positive integer that MUST divide n is 12.

Complete solution of this question is given above. Please ask if anything remains unclear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 15 Apr 2010
Posts: 45
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 02 Nov 2012, 03:35
Bunuel wrote:
Merging similar topics.

raviram80 wrote:
Hi All

I have a confusion about this question

169. If n is a positive integer and n2 is divisible by 72, then the largest positive integer that must divide n is
(A) 6
(8) 12
(C) 24
(0) 36
(E) 48

If we are looking for largest positive integer that must divide n, why can it not be 48.

because if n2 = 72 * 32 then n will be 48 , so does this not mean n is divisible by 48.

Please explain.

Thanks


The question asks about "the largest positive integer that MUST divide n", not COULD divide n. Since the least value of n for which n^2 is a multiple of 72 is 12 then the largest positive integer that MUST divide n is 12.

Complete solution of this question is given above. Please ask if anything remains unclear.


I spent a few hours on this one alone and I'm still not clear. I chose 12 at first, but then changed to 48.

I'm not a native speaker, so here is how I interpreted this question: "the largest positive integer that must divide n" = "the largest positive factor of n". Since n is a variable (i.e. n is moving), so is its largest factor. Please correct if I'm wrong here.

I know that if n = 12, n^2 = 144 = 2 * 72 (satisfy the condition). When n = 12, the largest factor of n is n itself, which is 12. Check: 12 is the largest positive number that must divide 12 --> true

However if n = 48, n^2 = 48 * 48 = 32 * 72 (satisfy the condition too). When n = 48, the largest factor of n is n itself, which is 48. Check: 48 is the largest positive number that must divide 48 --> true

So, I also notice that the keyword is "MUST", not "COULD". The question is, why is 48 not "MUST divide 48", but instead only "COULD divide 48"? I'm not clear right here. Why is 12 "MUST divide 12"? What's the difference between them?

Thanks,

Caten
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 02 Nov 2012, 03:53
8
catennacio wrote:
Bunuel wrote:
Merging similar topics.

raviram80 wrote:
Hi All

I have a confusion about this question

169. If n is a positive integer and n2 is divisible by 72, then the largest positive integer that must divide n is
(A) 6
(8) 12
(C) 24
(0) 36
(E) 48

If we are looking for largest positive integer that must divide n, why can it not be 48.

because if n2 = 72 * 32 then n will be 48 , so does this not mean n is divisible by 48.

Please explain.

Thanks


The question asks about "the largest positive integer that MUST divide n", not COULD divide n. Since the least value of n for which n^2 is a multiple of 72 is 12 then the largest positive integer that MUST divide n is 12.

Complete solution of this question is given above. Please ask if anything remains unclear.


I spent a few hours on this one alone and I'm still not clear. I chose 12 at first, but then changed to 48.

I'm not a native speaker, so here is how I interpreted this question: "the largest positive integer that must divide n" = "the largest positive factor of n". Since n is a variable (i.e. n is moving), so is its largest factor. Please correct if I'm wrong here.

I know that if n = 12, n^2 = 144 = 2 * 72 (satisfy the condition). When n = 12, the largest factor of n is n itself, which is 12. Check: 12 is the largest positive number that must divide 12 --> true

However if n = 48, n^2 = 48 * 48 = 32 * 72 (satisfy the condition too). When n = 48, the largest factor of n is n itself, which is 48. Check: 48 is the largest positive number that must divide 48 --> true

So, I also notice that the keyword is "MUST", not "COULD". The question is, why is 48 not "MUST divide 48", but instead only "COULD divide 48"? I'm not clear right here. Why is 12 "MUST divide 12"? What's the difference between them?

Thanks,

Caten


Only restriction we have on positive integer n is that n^2 is divisible by 72. The least value of n for which n^2 is divisible by 72 is 12, thus n must be divisible by 12 (n is in any case divisible by 12). For all other values of n, for which n^2 is divisible by 72, n will still be divisible by 12. This means that n is always divisible by 12 if n^2 is divisible by 72.

Now, ask yourself: if n=12, is n divisible by 48? No. So, n is not always divisible by 48.

Hope it's clear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
avatar
Joined: 10 Jan 2011
Posts: 158
Location: India
GMAT Date: 07-16-2012
GPA: 3.4
WE: Consulting (Consulting)
Reviews Badge
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 02 Nov 2012, 04:20
3
1
I approach this problem by prime factorisation.
any square must have 2 pairs of prime factors.
prime factorisation of 72 has 2*2, 3*3 and 2. n^2 must have one more 2 as a prime factor. Hence lasrgest number which must devide n is 2*3*2 = 12
_________________

-------Analyze why option A in SC wrong-------

Intern
Intern
avatar
Joined: 15 Apr 2010
Posts: 45
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 02 Nov 2012, 07:49
3
Bunuel wrote:

Only restriction we have on positive integer n is that n^2 is divisible by 72. The least value of n for which n^2 is divisible by 72 is 12, thus n must be divisible by 12 (n is in any case divisible by 12). For all other values of n, for which n^2 is divisible by 72, n will still be divisible by 12. This means that n is always divisible by 12 if n^2 is divisible by 72.

Now, ask yourself: if n=12, is n divisible by 48? No. So, n is not always divisible by 48.

Hope it's clear.


Thank you very much Bunuel. Very clear now. Now I understand what "must" means. It means it will be always true regardless of n. As you said (and I chose), when n = 24 or 36 or 48, the answer 48 can divide 48, but cannot divide 12. So the "must" here is not maintained. In this case we have to choose the largest factor of the least possible value of n to ensure that largest factor is also a factor of other values of n. Therefore the least value of n is 12, the largest factor of 12 is also 12. This factor also divides other n values, for all n such that n^2 = 72k.

My mistake was that I didn't understand the "must" wording and didn't check whether my answer 48 can divide ALL possible values of n, including n=12. This is what "must" mean.

Again, thanks so much!

Caten
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 02 Nov 2012, 07:53
1
catennacio wrote:
Bunuel wrote:

Only restriction we have on positive integer n is that n^2 is divisible by 72. The least value of n for which n^2 is divisible by 72 is 12, thus n must be divisible by 12 (n is in any case divisible by 12). For all other values of n, for which n^2 is divisible by 72, n will still be divisible by 12. This means that n is always divisible by 12 if n^2 is divisible by 72.

Now, ask yourself: if n=12, is n divisible by 48? No. So, n is not always divisible by 48.

Hope it's clear.


Thank you very much Bunuel. Very clear now. Now I understand what "must" means. It means it will be always true regardless of n. As you said (and I chose), when n = 24 or 36 or 48, the answer 48 can divide 48, but cannot divide 24 and 36. So the "must" here is not maintained. In this case we have to choose the largest factor of the least possible value of n to ensure that largest factor also a factor of other values of n. Therefore the least value of n is 12, the largest factor of 12 is also 12. This factor also divides other n values, for all n such that n^2 = 72k.

My mistake was that I didn't understand the "must" wording and didn't check whether my answer 48 can divide ALL possible values of n, including n=12. This is what "must" mean.

Again, thanks so much!

Caten


More must/could be true questions from our question banks (viewforumtags.php) here: search.php?search_id=tag&tag_id=193

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
avatar
Joined: 24 Mar 2013
Posts: 54
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 07 Mar 2014, 19:56
The explanations above are much clearer to me than the explanation offered in the official Gmat guide;

"Since 72k=(2^3)(3^2)k, then k=2m^2 for some positive integer m in order for 72k to be a perfect square."

Why must k=2m^2?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 08 Mar 2014, 06:08
2
actionj wrote:
The explanations above are much clearer to me than the explanation offered in the official Gmat guide;

"Since 72k=(2^3)(3^2)k, then k=2m^2 for some positive integer m in order for 72k to be a perfect square."

Why must k=2m^2?


A perfect square has its primes in even powers, thus k must complete odd power of 2 into even, hence 2, and it also can have some other integer in even power, hence m^2.

Hope it's clear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 50715
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 09 Mar 2014, 02:49
2
2
actionj wrote:
Cheers, that helped me nail down the fundamental concept I was missing.


Similar questions to practice:

if-n-is-a-positive-integer-and-n-2-is-divisible-by-96-then-127364.html
if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-90523.html
n-is-a-positive-integer-and-k-is-the-product-of-all-integer-104272.html
if-n-and-y-are-positive-integers-and-450y-n-92562.html
if-m-and-n-are-positive-integer-and-1800m-n3-what-is-108985.html
if-x-and-y-are-positive-integers-and-180x-y-100413.html
if-x-is-a-positive-integer-and-x-2-is-divisible-by-32-then-88388.html
if-5400mn-k-4-where-m-n-and-k-are-positive-integers-109284.html
if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-129929.html

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

e-GMAT Representative
User avatar
P
Joined: 04 Jan 2015
Posts: 2203
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 18 May 2015, 04:37
In questions like this,

It's a good idea to start with the prime factorized form of n.

We can write \(n = p1^m*p2^n*p3^r\). . . where p1, p2, p3 . . .are prime factors of n and m, n, r are non-negative integers (can be equal to 0)

So, \(n^2 = p1^{2m}*p2^{2n}*p3^{2r}\). . .

Now, \(n^2\) is completely divisible by 72 = \(2^3*3^2\)

This means, \(\frac{(p1^{2m}*p2^{2n}*p3^{2r} . . . )}{(2^3*3^2)}\) is an integer.

What does this tell you?

That p1 = 2 and 2m ≥ 3, that is m ≥ 3/2. But m is an integer. So, minimum possible value of m =2

Also, p2 = 3 and 2n ≥ 2. That is, n ≥ 1. So, minimum possible value of n = 1

Let's now apply this information on the expression for n:

n = \(2^2*3^1\)\(*something. . .\)

From this expression, it's clear that n MUST BE divisible by \(2^2*3^1\) = 12.

Takeaway: If you find yourself getting confused in questions that gives divisibility information about different powers of a number, start by writing a general prime factorized expression for the number raised to power 1. :)

Hope this was useful!

Japinder
_________________








Register for free sessions
Number Properties | Algebra |Quant Workshop

Success Stories
Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Must Read Articles
Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Manager
Manager
User avatar
B
Joined: 20 Jan 2017
Posts: 60
Location: United States (NY)
Schools: CBS '20 (A)
GMAT 1: 750 Q48 V44
GMAT 2: 610 Q34 V41
GPA: 3.92
Reviews Badge
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 25 Jan 2017, 08:27
1
1
1) n is squared. This means that it should have two identical sets of prime factors.
2) Since n^2 is divisible by 72, all prime factors of 72 should be prime factors of n^2.
3) The prime factors of 72 are 2*2*3*2*3. To make this product a perfect square we need to add one more 2. Then we get two identical sets of prime factors (2*2*3)=12.
4) n has to be at least 12 in order to satisfy the conditions of the problem. 12 is the largest integer that n MUST be divisible by.

The correct answer is B.

Posted from my mobile device
Target Test Prep Representative
User avatar
P
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 4170
Location: United States (CA)
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 09 Feb 2017, 16:24
4
amitvmane wrote:
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


We are given that n^2/72 = integer or (n^2)/(2^3)(3^2) = integer.

However, since n^2 is a perfect square, we need to make 72 or (2^3)(3^2) a perfect square. Since all perfect squares consist of unique primes, each raised to an even exponent, the smallest perfect square that divides into n^2 is (2^4)(3^2) = 144.

Since n^2/144 = integer, then n/12 = integer, and thus the largest positive integer that must divide n is 12.

Answer: B
_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Senior Manager
Senior Manager
User avatar
S
Joined: 08 Dec 2015
Posts: 294
GMAT 1: 600 Q44 V27
Reviews Badge
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 11 Feb 2017, 08:12
ScottTargetTestPrep

Thank you for your reply!

But I still dont get why the Q says the largest and not the smallest. 48 divides 12, so why not 48?
Target Test Prep Representative
User avatar
P
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 4170
Location: United States (CA)
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 14 Feb 2017, 15:57
3
iliavko wrote:
ScottTargetTestPrep

Thank you for your reply!

But I still dont get why the Q says the largest and not the smallest. 48 divides 12, so why not 48?


I think you may have misinterpreted the phrase "integer that must divide n." You interpreted it as “the integer must be divisible by n.” By your interpretation, if 12 is divisible by n, 48 is also divisible by n; this would be correct, had the wording of the question been as you interpreted it.

The phrase "integer that must divide n" really means “n must be divisible by that integer.” So if n is divisible by 12 (which means n/12 = integer), it doesn't mean n is divisible by 48 (i.e., it doesn't mean n/48 will be an integer). For example, if n = 12, n is divisible by 12, but n is not divisible by 48.

And thus, since we determined in the original question that n is a multiple of 12, n could be as small as 12, and the largest integer that must divide into 12 is 12. Does that answer your question?
_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Senior Manager
Senior Manager
User avatar
S
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 496
Location: India
Premium Member
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 19 Mar 2017, 23:39
asyahamed wrote:
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 = 3^2 x 2^3
since n^2 has a factor 2^3 we can say that 2^4 will also be a factor of this and 3^2 will be the factor of this

therefore n^2 has a definite factor, which is 144

n will be having a definite factor of 12.

Option B
_________________

GMAT Mentors
Image

CEO
CEO
User avatar
D
Joined: 11 Sep 2015
Posts: 3127
Location: Canada
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

Show Tags

New post 16 Nov 2017, 14:13
Top Contributor
2
amitvmane wrote:
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

---------------ASIDE #1--------------------------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:

If N is a factor by k, then k is "hiding" within the prime factorization of N

Consider these examples:
3 is a factor of 24, because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a factor of 70 because 70 = (2)(5)(7)
And 8 is a factor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a factor of 630 because 630 = (2)(3)(3)(5)(7)

---------------ASIDE #2--------------------------------------
IMPORTANT CONCEPT: The prime factorization of a perfect square will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even number of primes allows us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2x2x2x2x2x2x3x3 = (2x2x2x3)(2x2x2x3) = (2x2x2x3)²
--------NOW ONTO THE QUESTION!------------------

Given: n² is divisible by 72 (in other words, there's a 72 hiding in the prime factorization of n²)
So, n² = (2)(2)(2)(3)(3)(?)(?)(?)(?)(?)... [the ?'s represent other possible primes in the prime factorization of n²]
Since we have an ODD number of 2's in the prime factorization, we can be certain that there is at least one more 2 in the prime factorization.
So, we know that n² = (2)(2)(2)(3)(3)(2)(?)(?)(?)(?)
So, while there MIGHT be tons of other values in the above prime factorization, we do know that there MUST BE at least four 2's and two 3's.
Now do some grouping to get: n² = [(2)(2)(3)(?)(?)...][(2)(2)(3)(?)(?)...]
From this we can see that n = (2)(2)(3)(?)(?)...

Question: What is the largest positive integer that must divide n?
(2)(2)(3) = 12.
So, the largest positive integer that must divide n is 12

Cheers,
Brent
_________________

Test confidently with gmatprepnow.com
Image

GMAT Club Bot
Re: If n is a positive integer and n^2 is divisible by 72, then &nbs [#permalink] 16 Nov 2017, 14:13

Go to page    1   2    Next  [ 24 posts ] 

Display posts from previous: Sort by

If n is a positive integer and n^2 is divisible by 72, then

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.