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

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

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31 Mar 2012, 02:36
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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

Can anyone explain it in very simple manner ?
[Reveal] Spoiler: OA
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31 Mar 2012, 02:47
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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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Quantitative Review 2nd Edition Ques [#permalink]

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21 Apr 2012, 14:04
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: Quantitative Review 2nd Edition Ques [#permalink]

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21 Apr 2012, 14:14
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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 [#permalink]

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14 Aug 2012, 22:50
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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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Re: Quantitative Review 2nd Edition Ques [#permalink]

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02 Nov 2012, 04:35
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: Quantitative Review 2nd Edition Ques [#permalink]

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02 Nov 2012, 04:53
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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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Re: If n is a positive integer and n^2 is divisible by 72, then [#permalink]

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02 Nov 2012, 05:20
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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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Re: Quantitative Review 2nd Edition Ques [#permalink]

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02 Nov 2012, 08:49
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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Re: Quantitative Review 2nd Edition Ques [#permalink]

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02 Nov 2012, 08:53
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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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11 Jan 2013, 23:31
taurean 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

[Reveal] Spoiler:
after some discussion

IMO B.

72 prime factors are $$2^3 * 3^2$$

$$n^2$$ divisible by 72 is 144 ==> ($$2^4 * 3^2$$).
So n is 12

largest positive integer divides 12 must be 12 itself

pls correct me if im wrong
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Last edited by shanmugamgsn on 12 Jan 2013, 00:43, edited 1 time in total.
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11 Jan 2013, 23:34
shanmugamgsn wrote:
taurean 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

[Reveal] Spoiler:
after some discussion

IMO B.

72 prime factors are $$2^2 * 3^2$$

$$n^2$$ divisible by 72 is 144 ==> ($$2^3 * 3^2$$).
So n is 12

largest positive integer divides 12 must be 12 itself

pls correct me if im wrong

Correction: Prime factorization for 72 is $$2^3 * 3^2$$ -- there are 3 twos.
Also your solution needs further reasoning. 12 is not the only number n can take.
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12 Jan 2013, 00:45
PraPon wrote:
shanmugamgsn wrote:
taurean 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

[Reveal] Spoiler:
after some discussion

IMO B.

72 prime factors are $$2^2 * 3^2$$

$$n^2$$ divisible by 72 is 144 ==> ($$2^3 * 3^2$$).
So n is 12

largest positive integer divides 12 must be 12 itself

pls correct me if im wrong

Correction: Prime factorization for 72 is $$2^3 * 3^2$$ -- there are 3 twos.
Also your solution needs further reasoning. 12 is not the only number n can take.

ya it was a typo.. I corrected it...
Why 12 cannot be solution here?

Ya only 12 is not number greater than and divisible by 72, but it is first number ! So i choose it
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Re: If n is a positive integer and n^2 is divisible by 72, then [#permalink]

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12 Jan 2013, 06:35
The language used in this problem is one of the things that might get you under timed conditions; "the greatest... that must" is the LEAST, just like Bunuel explained and it is very important to have this sort of Gmat wording clear in mind.

Other than that it's not too difficult.
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Re: Quant Review 2E -- Q#169 [#permalink]

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11 Aug 2013, 08:13
anshuman09 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

Firstly , is there any specific reason for contesting 48 as an answer?

Nonetheless, if$$n^2$$ is divisible by 72, then n must have a prime factorization of the kind : $$2^2*3*.....(more primes)$$. Now the power of 2 has to be atleast 2, because 72 contains $$2^3$$. Thus, n = 12*some primes. Thus, 12 is the largest positive integer which WILL/MUST divide n.

Hope this helps.
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Re: Quant Review 2E -- Q#169 [#permalink]

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11 Aug 2013, 11:55
mau5 wrote:
anshuman09 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

Firstly , is there any specific reason for contesting 48 as an answer?

Nonetheless, if$$n^2$$ is divisible by 72, then n must have a prime factorization of the kind : $$2^2*3*.....(more primes)$$. Now the power of 2 has to be atleast 2, because 72 contains $$2^3$$. Thus, n = 12*some primes. Thus, 12 is the largest positive integer which WILL/MUST divide n.

Hope this helps.

But If we choose the value of n as 48 then n^2 (48 * 48) will also be divisible by 72. As the same thing is happening in case of n=12, as you have explained.
Since we have to choose the largest value of n, why cant 48 be the right value of n?
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Re: Quant Review 2E -- Q#169 [#permalink]

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11 Aug 2013, 12:12
anshuman09 wrote:
mau5 wrote:
anshuman09 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

Firstly , is there any specific reason for contesting 48 as an answer?

Nonetheless, if$$n^2$$ is divisible by 72, then n must have a prime factorization of the kind : $$2^2*3*.....(more primes)$$. Now the power of 2 has to be atleast 2, because 72 contains $$2^3$$. Thus, n = 12*some primes. Thus, 12 is the largest positive integer which WILL/MUST divide n.

Hope this helps.

But If we choose the value of n as 48 then n^2 (48 * 48) will also be divisible by 72. As the same thing is happening in case of n=12, as you have explained.
Since we have to choose the largest value of n, why cant 48 be the right value of n?

Check here: if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-129929.html#p1067773
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Re: Quant Review 2E -- Q#169 [#permalink]

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11 Aug 2013, 12:17
Thanks a lot Bunuel.
Now I got it.
My understanding is:

possible values of n = 12, 24, 36 and 48.
But it is 12 that can divide all the possible values of n. If we consider 48, it will not divide 12, 24 and 36.
Hence, 12 is the value that MUST divide n.

Thanks again.
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Re: Quant Review 2E -- Q#169 [#permalink]

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12 Aug 2013, 15:36
anshuman09 wrote:
mau5 wrote:
anshuman09 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

Firstly , is there any specific reason for contesting 48 as an answer?

Nonetheless, if$$n^2$$ is divisible by 72, then n must have a prime factorization of the kind : $$2^2*3*.....(more primes)$$. Now the power of 2 has to be atleast 2, because 72 contains $$2^3$$. Thus, n = 12*some primes. Thus, 12 is the largest positive integer which WILL/MUST divide n.

Hope this helps.

But If we choose the value of n as 48 then n^2 (48 * 48) will also be divisible by 72. As the same thing is happening in case of n=12, as you have explained.
Since we have to choose the largest value of n, why cant 48 be the right value of n?

12 must divide n , 24,36,48 may divide n .
n^2 / 72, here n can be 12, 36,48.......
but to be divided by 72, n should be at least 12. (12^2 = 144)
so 12 is must and the rest are may or could............
below 12 not possible. At least 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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13 Aug 2013, 10:32
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n^2/72 --> n*n/(2^3*3^2) -->(n/2*3)*(n/2*3)*1/2 --> means that n/6*1/2 --> 12
Re: If n is a positive integer and n^2 is divisible by 72, then   [#permalink] 13 Aug 2013, 10:32

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