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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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08 Feb 2019, 04:18
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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

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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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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

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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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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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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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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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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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