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(\frac{2}{3})^x  an integer?

A. 3

B. 4

C. 5

D. 8

E. 9

Princ · Accepted Answer

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

spc11 · Answer

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

Anki2609 · Answer

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: E

sumitkrocks · Answer

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

Shobhit7 · Answer

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

SSM700plus · Answer

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.

sudarshan22 · Answer

+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  .

PKN · Answer

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

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