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For how many integer values of x is 864(2/3)^x an integer?
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19 May 2018, 08:37
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Re: For how many integer values of x is 864(2/3)^x an integer?
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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?
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19 May 2018, 09:04
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?
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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?
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19 May 2018, 09:19
Let us write factors of the given term ..864 *(2/3)^x = 2^(5+x) *8 ^(3x) For it to be integer ..powers of prime factors need to be greater than or equal to zero.. So, 5+x>=0 and 3x>=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?
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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?
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19 May 2018, 10:55
souvik101990 wrote: 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?
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10 Aug 2018, 02:59
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?
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10 Aug 2018, 03:40
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 \(864(\frac{2}{3})^x\)=\(2^5*3^3*(\frac{2}{3})^x\)=\(2^{5+x}*3^{3x}\) So , the above expression is an integer when x={5,4,.......,0,............,3} . Total 9 integer values. Ans. (E)
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