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30 Dec 2023, 22:47
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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
A. 3
B. 4
C. 7
D. 8
E. 9
30 Dec 2023, 22:47
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Official Solution:
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 \(2,160*(\frac{2}{3})^k\) is an integer, we need to consider how the factors of 2,160 interact with \((\frac{2}{3})^k\).
When k is positive, \((\frac{2}{3})^k\) 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, \((\frac{2}{3})^k\) 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
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 \(2,160*(\frac{2}{3})^k\) is an integer, we need to consider how the factors of 2,160 interact with \((\frac{2}{3})^k\).
When k is positive, \((\frac{2}{3})^k\) 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, \((\frac{2}{3})^k\) 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
