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

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

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

Hope it's helps.
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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23
5
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.
##### General Discussion
Manager  Joined: 09 Jul 2010
Posts: 71
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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4
Hi All

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.

Thanks
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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4
1
Merging similar topics.

raviram80 wrote:
Hi All

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.

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.
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Intern  Joined: 15 Apr 2010
Posts: 36
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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1
Bunuel wrote:
Merging similar topics.

raviram80 wrote:
Hi All

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.

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 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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12
3
catennacio wrote:
Bunuel wrote:
Merging similar topics.

raviram80 wrote:
Hi All

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.

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.
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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8
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
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Intern  Joined: 15 Apr 2010
Posts: 36
Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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4
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 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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1
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.
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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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 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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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.
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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

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4
1
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
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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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.

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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

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.

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

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ScottTargetTestPrep

But I still dont get why the Q says the largest and not the smallest. 48 divides 12, so why not 48?
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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iliavko wrote:
ScottTargetTestPrep

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?
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Re: If n is a positive integer and n^2 is divisible by 72, then  [#permalink]

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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
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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

---------------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
_________________ Re: If n is a positive integer and n^2 is divisible by 72, then   [#permalink] 16 Nov 2017, 14:13

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