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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]

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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]

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12 Dec 2012, 03:02
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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

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

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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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14 Dec 2012, 02:20
Ans:

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]

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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]

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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.

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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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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 [ 20.75 KiB | Viewed 13246 times ]

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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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11 Sep 2014, 01:40
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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11 Sep 2014, 10:29
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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

72 = 2*2*2*3*3

72/ 2^k = int ; 72/2^k+1 not integer

if k= 2 then divisible ,k=3 then also ..
if K=3 then divisible , k=4 then not..

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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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02 Mar 2015, 01:35
If we say that 2^k is a divisor of 72 could we also do this to solve the problem?

Do the prime factorization of 72: 2^3 * 3^3, and then create the following equation:

2^k * 3^3 = 2^3 * 3^3, so k = 3?

Then we are assuming that k is 72, but 72 is still a divisor of 72. So, could we also do that?

As I see it, it is just a more visual way to say what everyone said above. Right...?
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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28 Aug 2015, 20:28
Bunuel 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.

Why is the answer not A? if k= 2 it is still a divisor of 72
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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29 Aug 2015, 01:16
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harishbiyani8888 wrote:
Bunuel 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.

Why is the answer not A? if k= 2 it is still a divisor of 72

We need such k that 2^k IS a divisor of 72 but 2^(k+1) is NOT. k cannot be 2 because 2^(2+1) IS a divisor of 72.
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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29 Aug 2015, 01:52
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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14 Mar 2016, 07:18
Great Question..
Here only K=3 satisfies the condition such that 8 divides 72 and 16 does not
Hence B
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Re: For the positive integers a, b, and k, a^k||b means that a^k [#permalink]

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18 May 2016, 11:30
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

Solution:

This is called a "defined function" problem. The parallel lines mean intrinsically nothing, except to establish a relationship between a^k and b.We are given that a^k || b means:

1) b/a^k = integer

2) b/a^(k+1) ≠ integer

Next we are given specific numbers 2^k || 72, and we must use the pattern to determine k, using a = 2 and b = 72; thus, we know:

72/2^k = integer AND 72/2^(k+1) ≠ integer

In order for 72/2^k = integer AND 72/2^(k+1) ≠ integer to be true, k must equal 3. If have trouble seeing how this works, we can plug 3 back in to prove it.

When k = 3, we know:

1) 72/2^3 = 72/8 = 9, which IS an integer.

AND

2) 72/2^(3+1) = 72/2^4 = 72/16 = 4 1/2, which is NOT an integer.

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]

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06 Jun 2016, 07:42
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

2^k ||72 means 2^k is a divisor of 72, but 2^(k+1) is not a divisor of 72

72 = 2^3*3^2

The maximum powers of 2 in 72 = 3
Hence 2^3 is a divisor of 72 and 2^4 is not a divisor of 72
k = 3

Correct Option: B
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Re: For the positive integers a, b, and k, a^k||b means that a^k   [#permalink] 06 Jun 2016, 07:42
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# For the positive integers a, b, and k, a^k||b means that a^k

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