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For the positive integers a, b, and k, a^k||b means that a^k

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For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 12 Dec 2012, 02:59
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For the positive integers a, b, and k, a^k||b means that a^k is a divisor of b, but a^(k + 1) is not a divisor of b. If k is a positive integer and 2^k||72, then k is equal to

(A) 2
(B) 3
(C) 4
(D) 8
(E) 18
[Reveal] Spoiler: OA
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 12 Dec 2012, 03:02
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Walkabout wrote:
For the positive integers a, b, and k, a^k||b means that a^k is a divisor of b, but a^(k + 1) is not a divisor of b. If k is a positive integer and 2^k||72, then k is equal to

(A) 2
(B) 3
(C) 4
(D) 8
(E) 18


72=2^3*3^2, so we have that 2^3 is a divisor of 72 and 2^4 is not. Thus 2^3||72, hence k=3.

Answer: B.
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 14 Dec 2012, 02:20
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72= 2^3x3^2, since 2^3 is a divisor of 72 k can be 3. Also 2^4=(2^(k+1)) is not a divisor of 72 , therefore the answer is (B).
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 14 Dec 2012, 22:44
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Initially looking to the problem one may try to plugin the numbers one by one. Here, 2^2=4 is a divisor of 72 and 2^3=8 is also a divisor of 72. But, we have to choose only one answer.
72=2x2x2x3x3=2^3 *3^2 and it is given that 72/2^k = integer. Here, we can equate 2^3=2^k and hence k=3. But, in fact 2^2 is also a divisor of 72 hence 2 could also be the answer. But, since it is additionally given that k+1 is not a divisor i.e. 2 in this case does not satisfy the condition because 2+1=3 and 2^3 is a divisior of 72. where as 3 satisfies the condition i.e. 3+1= 4 turning into 2^4 which is not a divisor of 72. This is how only one answer choice is left which is equal to 3 = answer choice B.
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 19 Dec 2012, 06:15
2^k has to be a factor of 72.

Factors of 72: 3^2 x 2^3

Hence, k = 3.

Answer B.
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink] New post 25 Jul 2013, 07:30
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Prime factorization is also helpful to solve this problem. For ME it´s faster..

Probably it helps someone.
Attachments

PS 110 - prime factorization.jpg
PS 110 - prime factorization.jpg [ 20.75 KiB | Viewed 2656 times ]

Re: For the positive integers a, b, and k, a^k||b means that a^k   [#permalink] 25 Jul 2013, 07:30
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