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# If n is a positive integer and n2 is dividible by 72, then

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Manager
Joined: 28 Jan 2005
Posts: 71
If n is a positive integer and n2 is dividible by 72, then  [#permalink]

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16 Feb 2005, 17:12
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100% (00:24) correct 0% (00:00) wrong based on 24 sessions

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If n is a positive integer and n2 is dividible by 72, then largest positive integer that must divide n is

A : 6
B : 12
C : 24
D : 36
E : 48

OG PS #412
----------------------------

-----------------------------

The process I took :
n2 = 72 x k(for some positive integer) = (2 x 3)2 x 2 x k

1) if k = 2
n2 = (12)2 Then, 12 is the maximum number which can divide n.

2) if k = 8
n2 = (24)2 Then, 24 is the maximum number which can divide n.

3) if k = 18
n2 = (36)s Then, 36 is the maximum number which can divide n.

If my study above is correct, why is the correct answer B(n=12)? As the above, n can be 12, 24, 36 or more!

Can you someone advise me which I made a wrong understanding? Thank you.

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VP
Joined: 25 Nov 2004
Posts: 1450
Re: OG PS No. 412  [#permalink]

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16 Feb 2005, 19:49
1
1
if n^2 (nxn) is divided by 72 (i.e 6X6X2), n^2 is also divided by 144(72X2) because n^2 is a square and its dividors must be squars. therefore, the factors of n^2 must be at least 2X2X2X2X3X3 (12). the sqrt of n^2 is n, which must be divided by 12, sqrt of 144.

explain later, if unclear.
VP
Joined: 13 Jun 2004
Posts: 1097
Location: London, UK
Schools: Tuck'08

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16 Feb 2005, 20:51
Is the question really good because in my mind there are other possible answers but those answers are not in the proposed choices a,b,c,d or e.

Because 144 could also be the answer, no ?
n=144, n^2 = 20736 it is divisble by 72

Is backsolving the only way or am I just missing something here ?
Manager
Joined: 28 Jan 2005
Posts: 71
Re: OG PS No. 412  [#permalink]

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18 Feb 2005, 22:37
1
Dear MA,
Thank you for your kind support. I have just one question to your responses below. I see your point that the factors of n^2 must be at least 2X2X2X2X3X3 (12).

However, what do you think of " the factors of n^2 are for instance 2X2X2X2X2X2X3X3 (24)? Because the question is asking the "largest" positive number that must divide n, I'm wondering why 12 is the "largest" number. As shown, 24 can also devide n and 576(24^2) can be divided by 72(576 / 72 = 8).

Can you clarify my question, please? Thank you.

MA wrote:
if n^2 (nxn) is divided by 72 (i.e 6X6X2), n^2 is also divided by 144(72X2) because n^2 is a square and its dividors must be squars. therefore, the factors of n^2 must be at least 2X2X2X2X3X3 (12). the sqrt of n^2 is n, which must be divided by 12, sqrt of 144.

explain later, if unclear.

_________________

Best regards,

Senior Manager
Joined: 02 Feb 2004
Posts: 341

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19 Feb 2005, 12:58
quick solution:

72=2 x 6x6
n^2=nxn

therefore for n^2 to be divisable by 72, one of the n out of the two must be divisable by 12.
Manager
Joined: 28 Jan 2005
Posts: 71

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19 Feb 2005, 22:27
Dear mirhaque,

n^2 = 72 x k = 2x6x6xk

If k = 2, yes, as you explained, n^2 = 12^2 and then, n = 12

However,

If k = 8 = 2^3, n^2 = (2x2x2x3)^2 = 24^2 and then, n = 24

Can you please point out my mistake? Thank you.

mirhaque wrote:
quick solution:

72=2 x 6x6
n^2=nxn

therefore for n^2 to be divisable by 72, one of the n out of the two must be divisable by 12.

_________________

Best regards,

VP
Joined: 25 Nov 2004
Posts: 1450
Re: OG PS No. 412  [#permalink]

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19 Feb 2005, 23:14
Taku wrote:
what do you think of " the factors of n^2 are for instance 2X2X2X2X2X2X3X3 (24)? Because the question is asking the "largest" positive number that must divide n, I'm wondering why 12 is the "largest" number. As shown, 24 can also devide n and 576(24^2) can be divided by 72(576 / 72 = 8). Can you clarify my question, please? Thank you.

taku,
n^2 is divisible by 72 means one n is divisible by 12 (6x2=2x2x3) and another n is divisible by 6 (2x3) and another number, lets say.

n^2/72=nxn/2x2x2x3x3=n/(2x2x3) x n/(2x3xk). k must be 2 because n/(2x2x3)=n/(2x3xk)

your question why k can not be 8? k can not be 8 because n^2 is divisible by 72 means it is also be divisible by 144 because factor of n^2 must have square factors, which 144 fullfils. if k =8, then 288, instead of 72, would be the divisor. but if n=12 and n^2 = 144, does n^2 (144) divided by 288. no it is not possible. therefore, it is only 12 the largest positive integer that divides n under the given circumstances.
Manager
Joined: 28 Jan 2005
Posts: 71
Re: OG PS No. 412  [#permalink]

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23 Feb 2005, 04:32
Dear MA,
First of all, thank you very much for your support in this regards. Really appreciate your kind attention. Sorry but can I ask just one question about your responses.

You are stating that if k =8, then 288, instead of 72, would be the divisor. Can you tell me why "288"? Of course, I see n^2=(24)^2=576 which can be divided by 288(576 / 2 = 288). Is this why you are mentioning "288"? Sorry I don't see the reason or background of "288". Can you explain it? Thank you very much.

MA wrote:
Taku wrote:
what do you think of " the factors of n^2 are for instance 2X2X2X2X2X2X3X3 (24)? Because the question is asking the "largest" positive number that must divide n, I'm wondering why 12 is the "largest" number. As shown, 24 can also devide n and 576(24^2) can be divided by 72(576 / 72 = 8). Can you clarify my question, please? Thank you.

taku,
n^2 is divisible by 72 means one n is divisible by 12 (6x2=2x2x3) and another n is divisible by 6 (2x3) and another number, lets say.

n^2/72=nxn/2x2x2x3x3=n/(2x2x3) x n/(2x3xk). k must be 2 because n/(2x2x3)=n/(2x3xk)

your question why k can not be 8? k can not be 8 because n^2 is divisible by 72 means it is also be divisible by 144 because factor of n^2 must have square factors, which 144 fullfils. if k =8, then 288, instead of 72, would be the divisor. but if n=12 and n^2 = 144, does n^2 (144) divided by 288. no it is not possible. therefore, it is only 12 the largest positive integer that divides n under the given circumstances.

_________________

Best regards,

SVP
Joined: 03 Jan 2005
Posts: 2170
Re: OG PS No. 412  [#permalink]

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23 Feb 2005, 08:14
Taku wrote:
If n is a positive integer and n2 is dividible by 72, then largest positive integer that must divide n is

The key word is "must". This means for each and every possible n, this integer must divide it.

We know that 2 must divide n, as well as 6, as well as 12. However, for some ns, eg. n=12, n[sup]2[/sup]=144 is divisible by 72, 18 cannot be a divisor. Therefore we have to choose the largest among 2, 3, 6 and 12. And the answer is 12.
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Re: If n is a positive integer and n2 is dividible by 72, then  [#permalink]

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31 Oct 2017, 18:59
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Re: If n is a positive integer and n2 is dividible by 72, then &nbs [#permalink] 31 Oct 2017, 18:59
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