12 Days of Christmas GMAT Competition - Day 2: For how many integer

Question

12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun

For how many integer values of k is  2,160*(\frac{2}{3})^k  an integer?

A. 3
B. 4
C. 7
D. 8
E. 9

Bunuel · Accepted Answer

GMAT Club Official Explanation:

For how many integer values of k is  2,160*(\frac{2}{3})^k  an integer?

A. 3
B. 4
C. 7
D. 8
E. 9

First, let's factorize 2,160:

2,160 = 2^4 * 3^3 * 5

To determine when 2,160*(2/3)^k is an integer, we need to consider how the factors of 2,160 interact with (2/3)^k.

When k is positive, (2/3)^k introduces 3^k in the denominator. For the expression to stay an integer, the 3s in the denominator must cancel out the 3s in 2,160's factorization. Thus, in this case k can take three values: 1, 2, and 3.

When k is negative, (2/3)^k introduces 2^|k| in the denominator. The 2s in the denominator must not exceed the four 2s in 2,160's factorization. Thus, in this case, k can take four values: -1, -2, -3, and -4.

Don't forget that k can also be 0, as any nonzero number to the power of 0 is 1, making the entire expression just 2,160.

So, the possible values for k are -4, -3, -2, -1, 0, 1, 2, 3.

Answer: D.

Kinshook · Answer

Asked: For how many integer values of k is \(2,160*(\frac{2}{3})^k\) an integer?

\(2,160*(\frac{2}{3})^k = 2^4*3^3*5 * 2^k * 3^{-k} = 2^{4+k}*3^{3-k}*5\)
Is an integer when
3-k>=0; k<=3
4+k>=0; k>=-4
-4<=k<=3
k = {-4,-3,-2,-1,0,1,2,3}: 8 integer values of k

IMO D

Suboopc · Answer

2160 can be broken down as 
2160 = 2^4 * 3^3 * 5

Since we are counting integers, we have to consider positive integers, negative integers and zero as well.

So, the values of k for which denominator of the fraction cancels out are
-4, -3, -2, -1, 0, 1, 2, 3

Answer = 8

missionmba2025 · Answer

Value of k can be positive, negative or zero.

Power of 3 in 2160 = 3

2160 / 3 = 720
710 / 3 = 240
240 / 3 = 80

Power of 2 in 2160 = 4

2160 / 2 = 1080
710 / 2 = 540
240 / 2 = 270
270 / 2 = 135

Total Powers = 3 + 4 + 1 = 8

IMO D

limuengaoey · Answer

For how many integer values of k is 2,160∗(2/3)^k

A. 3
B. 4
C. 7
D. 8
E. 9

2160 = (2^4)*(3^3)*5
hence K can be 0,1, 2,3
K can be -1,-2,-3,-4
Hence K can be 8 numbers -->D

treeetravel · Answer

for K=0, value comes 2160

K= 1, value comes 720*2

k=2, Value comes 240 *4

k= 3, value comes 80* 8

after that, for K >4, the value comes as fraction

parallelly for

k= -1, values comes 1080*3

k= -2, values comes 540*9

k= -3, values comes 270*27

k= -4, values comes 135*81

after that, for K <-4, the value comes as fraction

so total no. of K values = 8

rahumangal · Answer

12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun

For how many integer values of k is  2,160*(\frac{2}{3})^k  an integer?

A. 3
B. 4
C. 7
D. 8
E. 9

now k can be both positive and negative 
2160= 2^4*3^3*5
so values of k= -4,-3,-2,-1,0,1,2,3
total no of values =8

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12 Days of Christmas GMAT Competition - Day 2: For how many integer

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