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If 12^13/x is an integer, which of the following could be the value of

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If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post 08 Feb 2019, 04:18
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

56% (01:36) correct 44% (01:24) wrong based on 60 sessions

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Re: If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post 08 Feb 2019, 04:40
Bunuel wrote:
If 12^13/x is an integer, which of the following could be the value of x ?

I. 36
II. 64
III. 432

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


I hope the question is \(\frac{12^{13}}{x}\) is an Integer

i.e. x is a factor of \(12^{13} = 2^{26}*3^{13}\)

I. 36 =\(2^2*3^2\) which is a factor of \(2^{26}*3^{13}\)
II. 64 = \(2^6\) which is a factor of \(2^{26}*3^{13}\)
III. 432 = 8*54 = \(2^4*3^3\) which is a factor of \(2^{26}*3^{13}\)

Hence all given values can be values of x

Answer: Option E
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Re: If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post Updated on: 08 Feb 2019, 04:43
Bunuel wrote:
If 12^13/x is an integer, which of the following could be the value of x ?

I. 36
II. 64
III. 432

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


\(12^{13} is quite a big number .\)

suppose we are just dealing with \(12^2\)

\(12^2 = 144.\)

\(144 = 12 * 12 = 3^2 2^4.\)

I) \(36 = 3*12 = 3 * 3*4 = 3^2 2^2.\)

II) \(64 = 2^3 2^3. 2^{3 + 3} = 2^6.\)

144 gives us \(2^4\). If we get 2^4 from \(12^2\). \(12^13\) will surely yield \(2^6.\)

III) \(432 = 3 *144. = 3 * 12 * 12 = 3 * 3*4 * 3 * 4 = 3^3 2^2 2^2 = 3^3 2^4.\)

\(12^2 = 3^2 2^4.\) So, \(2^4\)matches. \(12^2\) yields \(3^2\) . \(12^13\) will surely give us 3^3.


****\(12 = 3*2^2\). If we consider the primes of 12, we can easily predict that all of the integers , which are the values of x, can divide into \(12^13\).

Originally posted by KSBGC on 08 Feb 2019, 04:43.
Last edited by KSBGC on 08 Feb 2019, 04:43, edited 1 time in total.
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If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post 08 Feb 2019, 04:43
Bunuel wrote:
If 12^13/x is an integer, which of the following could be the value of x ?

I. 36
II. 64
III. 432

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III


So lets see, if i got this part right then the question is

(12^13) / x

\((2^2 * 3)^{13}\)

\(2^{26} * 3^{13}\)

All 3 are sufficient

36 = 2*2*3*3
64 = \(2^6\)
432 = \(2^4 * 3^3\)
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If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post 08 Feb 2019, 05:00
Bunuel wrote:
If \(\frac{12^{13}}{x}\) is an integer, which of the following could be the value of x ?

I. 36
II. 64
III. 432

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III



Keyord: could be the value of x ?

We see 12*13/x

So anything that has mutiples 2's ,3's and one 13 can be a contender since they can be part of cancellation from Numerator

1.36 has 12*3 yes possible
2.64 has 4^3 which is definitely possible since 12^13 will yield (3*4)^13 so 13 powers of 4
3.432 again is definitely possible 432=4*4*3^3

Hence all three are sufficient!!


Hope it helps!!
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Re: If 12^13/x is an integer, which of the following could be the value of  [#permalink]

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New post 08 Feb 2019, 06:24
Bunuel wrote:
If \(\frac{12^{13}}{x}\) is an integer, which of the following could be the value of x ?

I. 36
II. 64
III. 432

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III



for \(\frac{12^{13}}{x}\) to be an integer , value of x must be a factor of m]\frac{12^{13}}{x}[/m]


which can be written as : 2^26 * 3^13

upon reviewing all given options ; we can say that all the no 36,64 & 432 would be give an integer value when replaced as x in m]\frac{12^{13}}{x}[/m]

IMO E
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Re: If 12^13/x is an integer, which of the following could be the value of   [#permalink] 08 Feb 2019, 06:24
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