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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If n is a positive integer and n^2 is divisible by 72, then

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Intern  B
Joined: 06 Oct 2017
Posts: 8
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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ScottTargetTestPrep wrote:
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.

Hey Scott,

I think we may have covered it buy why can't we make n=72 therefore n^2 = (72)(72) ... why are we trying to use the SMALLEST perfect square .. using n=72 follows the rules set out in the question... ???
Math Expert V
Joined: 02 Sep 2009
Posts: 64242
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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YYZ wrote:
ScottTargetTestPrep wrote:
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.

Hey Scott,

I think we may have covered it buy why can't we make n=72 therefore n^2 = (72)(72) ... why are we trying to use the SMALLEST perfect square .. using n=72 follows the rules set out in the question... ???

The question asks to find the largest positive integer that MUST divide n. So, which ALWAYS divides n, if n^2 is divisible by 72. Now, while n COULD be divisible by any integer, for example, by by 48, 72, 1,000,000, ... it MUST be divisible only by factors of 12. Why? Because the least value of n for which n^2 is divisible by 144 is 12.

Hope it's clear.
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Intern  B
Joined: 11 Dec 2016
Posts: 47
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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ScottTargetTestPrep wrote:
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

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, n/12 = integer, and thus the largest positive integer that must divide n is 12.

Hi ScottTargetTestPrep, really good explanation. There's just one thing that's confusing me. If the question had asked what would be smallest integer, how would we proceed about it then?

Thanks!
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10593
Location: United States (CA)
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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asfandabid wrote:
ScottTargetTestPrep wrote:
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

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, n/12 = integer, and thus the largest positive integer that must divide n is 12.

Hi ScottTargetTestPrep, really good explanation. There's just one thing that's confusing me. If the question had asked what would be smallest integer, how would we proceed about it then?

Thanks!

They will not ask that since the smallest positive integer that divides another number is always 1.
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

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Manager  B
Joined: 04 Dec 2015
Posts: 157
WE: Operations (Commercial Banking)
If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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1
Bunuel i have understood why and how 12 is the answer, but why are we assuming that in order to find largest integer that must divide n, we first need to find the least/minimum possible value of n and not the maximum possible value of n ???

if i take maximum possible value of n ie 48, then 48 will be the largest positive integer that will divide n as per the same logic..

Look forward to hearing from you

ScottTargetTestPrep

Originally posted by INSEADIESE on 14 May 2019, 14:51.
Last edited by INSEADIESE on 23 May 2019, 11:32, edited 1 time in total.
Senior Manager  P
Joined: 31 May 2017
Posts: 412
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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1
ScottTargetTestPrep wrote:

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?

ScottTargetTestPrep

I also chose 48 as my answer and was racking my brain on why i got it wrong. But the explanation given above made me understand where i went wrong. I also interpreted the phrase wrongly as you had mentioned.

Thanks much for the clarity.
_________________
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10593
Location: United States (CA)
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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ScottTargetTestPrep wrote:

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?

ScottTargetTestPrep

I also chose 48 as my answer and was racking my brain on why i got it wrong. But the explanation given above made me understand where i went wrong. I also interpreted the phrase wrongly as you had mentioned.

Thanks much for the clarity.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager  B
Joined: 04 Dec 2015
Posts: 157
WE: Operations (Commercial Banking)
If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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ScottTargetTestPrep Bunuel i have understood why and how 12 is the answer, but why are we assuming that in order to find largest integer that must divide n, we first need to find the least/minimum possible value of n and not the maximum possible value of n ???

if i take maximum possible value of n ie 48, then 48 will be the largest positive integer that will divide n as per the same logic..

Look forward to hearing from you

ScottTargetTestPrep
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10593
Location: United States (CA)
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

### Show Tags

ScottTargetTestPrep Bunuel i have understood why and how 12 is the answer, but why are we assuming that in order to find largest integer that must divide n, we first need to find the least/minimum possible value of n and not the maximum possible value of n ???

if i take maximum possible value of n ie 48, then 48 will be the largest positive integer that will divide n as per the same logic..

Look forward to hearing from you

ScottTargetTestPrep

First of all, there is no maximum possible value of n since n can be as large as possible. For example, if a value of n is 24, you can multiply it by any positive integer greater than 1 to make it even larger. Secondly, we are finding the largest number that must divide n,; let’s say that number is d. That is, no matter what the value of n is, d must divide into n. Therefore, we need to find the smallest value of n such that d will divide into it.
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager  G
Joined: 29 Oct 2015
Posts: 237
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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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

If n = 6 , 6^2 is not divisible by 72.

If n= 12 , 12^2 is divisible by 72. And 12 is the largest positive integer that must divide n.

If n = 24 , then also 24 ^ 2 is divisible by 72. But 24 is not largest positive integer that must divide n . Because n can be 12 also and 24 can not divide 12. If you divide 12 by 24 , it will leave a remainder of 12. So 24 can not be largest positive integer which must divide n.

If n = 36 , then also 36^2 is divisible by 72 . But 36 can not be largest positive integer that must divide n. Because n can be 12 also and 36 can not divide 12.

Same logic applies for 48.

Please give me kudo s if you liked my explanation.

Manager  S
Joined: 19 Jan 2018
Posts: 91
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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1
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 know that $$\frac{n^2}{72}$$ = Integer

We can break down 72 into its Prime numbers; $$\frac{n^2}{2^3 * 3^2}$$

$$n^2$$ must have even powers for its prime, while still being able to be divided into 72, so $$n^2 = 2^4 * 3^2$$

If $$n^2 = 2^4 * 3^2$$, then $$n = 2^2 * 3$$

$$n=12$$, so the largest positive integer than can divide into n is 12.

Intern  B
Joined: 06 Mar 2019
Posts: 46
Schools: IMD '21
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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When you say "the largest positive integer that must divide n", doesn't that mean the number is n? The largest number that can divide a number is itself. With that, I looked at the values and 12 made sense.

Bunuel , ScottTargetTestPrep , Is this a correct analysis?
Senior Manager  G
Joined: 21 Feb 2017
Posts: 478
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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@veritasprep did a youtube session on similar questions. Very helpful to clear the fundamental of such problems. Re: If n is a positive integer and n^2 is divisible by 72, then   [#permalink] 09 Apr 2020, 02:25

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