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

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Intern
Joined: 06 Aug 2007
Posts: 38
If n is a positive integer and n^2 is divisible by 72, then [#permalink]

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30 Sep 2007, 14:50
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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
VP
Joined: 08 Jun 2005
Posts: 1145

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30 Sep 2007, 15:00
since (n*n)/72 = integer and 72 = 2*3*2*2*3

then:

Then the largest positive integer that must divide n is:

2*3 = 6

2*3*2 = 12

VP
Joined: 09 Jul 2007
Posts: 1100
Location: London

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30 Sep 2007, 15:13
Kalyan 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

E.
n=48
48*48/72=12*4*12*4/72=144*16/72

so the max is 48
Intern
Joined: 06 Aug 2007
Posts: 38

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30 Sep 2007, 15:15
KillerSquirrel,

Why stop with 2*3*2=12?

Why cant it be 24, 36?

Maybe I'm missing something very obvious here?? :(
VP
Joined: 08 Jun 2005
Posts: 1145

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30 Sep 2007, 15:29
Those kind of questions tend to be a bit confusing so I would like to better explain my above answer:

We have to work with n and with 72 where 72 = 2*3*2*2*3.

since (n*n)/72 = integer

then (n*n)/2*3*2*2*3 = integer

n can be 2*3*2 since (2*3*2)*(2*3*2)/(2*3*2*2*3) = 2 = integer

so our answer (not n) has to be 12.

you could ask why our answer can't be 24 since 72 = 2*3*2*2*3.

and (2*3*2*2*3)*(2*3*2*2*3)/(2*3*2*2*3) = 72

and 72/24 = 3

well it can - but the question asks about the largest positive integer that must divide n:

and if n is 12 or 72 the answer is 12 but not 24. But if n=72 then our answer can be 6, 12, 24 or 36.

so the largest positive integer that must divide n is 12.

Director
Joined: 22 Aug 2007
Posts: 566

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30 Sep 2007, 16:46
KillerSquirrel wrote:
Those kind of questions tend to be a bit confusing so I would like to better explain my above answer:

We have to work with n and with 72 where 72 = 2*3*2*2*3.

since (n*n)/72 = integer

then (n*n)/2*3*2*2*3 = integer

n can be 2*3*2 since (2*3*2)*(2*3*2)/(2*3*2*2*3) = 2 = integer

so our answer (not n) has to be 12.

you could ask why our answer can't be 24 since 72 = 2*3*2*2*3.

and (2*3*2*2*3)*(2*3*2*2*3)/(2*3*2*2*3) = 72

and 72/24 = 3

well it can - but the question asks about the largest positive integer that must divide n:

and if n is 12 or 72 the answer is 12 but not 24. But if n=72 then our answer can be 6, 12, 24 or 36.

so the largest positive integer that must divide n is 12.

I actually missed this detail too,

Thank you, KS.
Senior Manager
Joined: 24 Jul 2007
Posts: 290

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01 Oct 2007, 12:54
n^2/(2^3*3^2) = k

=> n = sqrt(k*(2^3*3^2))
=> n = 6*sqrt(2k)
its given that n is an integer. That can only be possible if there is another integer p such that k= 2*p^2
=> n = 12*p

Hence, 12 must divide n
01 Oct 2007, 12:54
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