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

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

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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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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?
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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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$$\frac{n*n}{72}$$ = $$\frac{n*n}{2*2*2*3*3}$$
This means n should be at least 2*2*3. Otherwise $$n^2$$ won't be divisible by 72

What is the largest integer that can divide n?
Let's see, largest integer that can divide 2? 2
Largest integer that can divide 3? 3
Largest integer that can divide n is n itself

Time Taken 2:31
Difficulty level 650
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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

Given that, n is a positive integer and n^2 is divisible by 72 means that n^2 is a perfect square that is lowest multiple of 72.

Or, the lowest multiple of 72 that is a perfect square is 144.

So, n^2 = 144q, q is some positive integer => 12 is a factor of n.

Originally posted by arunspanda on 18 Mar 2014, 08:00.
Last edited by arunspanda on 23 Mar 2014, 20:01, edited 2 times in total.
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Re: If n is a positive integer and n^2 is divisible by 72, then the larges [#permalink]
Bunuel wrote:
SOLUTION

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 questions to practice:

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Hi,

How can the part highlighted in bold be concluded fro the question stem "The largest positive integer that must divide n, means for the least value of n"??

In absence of this even 48 looks like a valid answer, as 48^2 is divisible by 72 and 48 is divisible by itself.

Regards,
Sagar
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itsworththepain wrote:
Bunuel wrote:
SOLUTION

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 questions to practice:

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if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-90523.html
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if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-129929.html

Hi,

How can the part highlighted in bold be concluded fro the question stem "The largest positive integer that must divide n, means for the least value of n"??

In absence of this even 48 looks like a valid answer, as 48^2 is divisible by 72 and 48 is divisible by itself.

Regards,
Sagar

We need the largest positive integer that MUST divide n. So, we should find the least value of n for which n^2 is divisible by 72, and if that n is divisible by some number then so will be every other n's (MUST condition will be satisfied). The least positive n for which n^2 is divisible by 72 is 12 (12^2 = 144, which is divisible by 72). So, even if n = 12 it is divisible by 12 but not divisible by 24, 36 or 48.

Follow the links in my previous post for similar questions to understand the concept better.
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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!

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

CONCEPT: Perfect Squares always have even power of Prime no. in their Prime factorised form.

$$72 = 8*9 = 2^3*3^2$$

Since power of prime numbers must be prime so $$72$$must be multiplied by minimum $$2^1$$ to make the result a perfect Square

In that Case $$n^2 = 2^4 * 3^2$$

i.e. $$n = 2^2 * 3 = 12$$