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For how many integer values of x is 864(2/3)^x an integer?

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For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 08:37
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E

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For how many integer values of x is \(864(\frac{2}{3})^x\) an integer?


A. 3

B. 4

C. 5

D. 8

E. 9

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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post Updated on: 19 May 2018, 09:11
Correcting myself:
For 864 * (2/3)^x to be an integer, x should be less than or equal to the power of 3 or 2 in 864.
Thus, 864 = 32*3^3

2^5 and 3^ 3 gives us: -5, -4, -3, -2, -1, 0, 1, 2, 3
Answer E

Originally posted by spc11 on 19 May 2018, 08:57.
Last edited by spc11 on 19 May 2018, 09:11, edited 2 times in total.
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 09:04
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OA:E
\(864 = 2^5*3^3\)\(\)
So integer values of x such that expression \(864(\frac{2}{3})^x\)an integer can be \(-5,-4,-3,-2,-1,0,1,2,3\)

x can have total 9 different integer value
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 09:17
864 (2/3)

Prime factorizing 864 = \(2^5\) * \(3^3\)
so the value of m can be anything between -5 to 3

total 9 values

Answer:
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 09:19
Let us write factors of the given term ..864 *(2/3)^x = 2^(5+x) *8 ^(3-x)
For it to be integer ..powers of prime factors need to be greater than or equal to zero..
So, 5+x>=0 and 3-x>=0
hence we get -5 <=X<=3 i.e integral value X can have ..-5,-4,-3,-2,-1,0,1,2,3 ...i.e 9

Answer is E
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 09:45
864= 27*32= 3^3*2^5

Hence,

* If x is non negative Integer, 3^x is the denominator, so x can have four different values:
3^0, 3^1, 3^2 and 3^3.


* If x is a negative Integer, then 2^x will become the denominator, so x can have values from -5 to -1

So, in total, x can have values from -5 to +3= 9 values

Ans E
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 19 May 2018, 10:55
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souvik101990 wrote:

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For how many integer values of x is \(864\)\((2/3)\)\(^x\) an integer?

A. 3

B. 4

C. 5

D. 8

E. 9


The question is to find out how many different values can x take . And \(864\)\((2/3)\)\(^x\) should be an integer.

\(864 = 2^5 * 3^3\)

there are three possibilities.
1) x is positive

If x is positive. x can take maximum of 3 values only. x = 1, x=2, x=3.
When x =1, One 3 in denominator cancels one of the three in 864.
When x =2, Two 3's in denominator cancels two of the three in 864.
When x =3, Three 3's in denominator cancels all 3's in 864.

2) x is negative.

If x is negative. The expression will change two \(864\)\((3/2)\)\(^x\)
x can take maximum of 5 values only. x = -1, x=-2, x=-3, x=-4, x=-5 .
Similarly
When x =-1, One 2 in denominator cancels one of the Two in 864.
When x =-2, Two 2's in denominator cancels two of the two in 864.
When x =-3, Three 2's in denominator cancels three 2's in 864.
When x =-4, Four 2's in denominator cancels four 2's in 864.
When x =-5, Five 2's in denominator cancels five 2's in 864.

3) x = 0. The expression will be integer.

Total 9 different values are possible.

Ans - E.

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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 10 Aug 2018, 02:59
1
Bunuel wrote:
For how many integer values of x is \(864(\frac{2}{3})^x\) an integer?

+1 for E.

864 = 2^5 ∗ 3^3
So, every value of x needs to deduct a power of either 2 or 3 for the value as a whole to be integer.

Now, 864 * ( 2 / 3 )^x = ( 2^5 ∗ 3^3 ) * ( 2 / 3 )^x
x can be −5, −4, −3, −2, −1, 0, 1, 2, 3 (in total 9) for the value as a whole to be integer.



Hence, E.
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Re: For how many integer values of x is 864(2/3)^x an integer?  [#permalink]

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New post 10 Aug 2018, 03:40
1
Bunuel wrote:
For how many integer values of x is \(864(\frac{2}{3})^x\) an integer?

A. 3
B. 4
C. 5
D. 8
E. 9


\(864(\frac{2}{3})^x\)=\(2^5*3^3*(\frac{2}{3})^x\)=\(2^{5+x}*3^{3-x}\)

So , the above expression is an integer when x={-5,-4,.......,0,............,3} . Total 9 integer values.

Ans. (E)
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Re: For how many integer values of x is 864(2/3)^x an integer? &nbs [#permalink] 10 Aug 2018, 03:40
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