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Manager  Joined: 02 Dec 2012
Posts: 173
For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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72 00:00

Difficulty:   25% (medium)

Question Stats: 75% (01:44) correct 25% (02:00) wrong based on 2616 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58320
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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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Intern  Joined: 20 Aug 2012
Posts: 3
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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3
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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Intern  Joined: 24 Apr 2012
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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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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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2^k has to be a factor of 72.

Factors of 72: 3^2 x 2^3

Hence, k = 3.

Intern  Joined: 18 May 2013
Posts: 7
Location: Germany
GMAT Date: 09-27-2013
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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3
2
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 38327 times ]

Manager  Joined: 07 Apr 2014
Posts: 101
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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2
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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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...?
Intern  Joined: 12 Nov 2013
Posts: 39
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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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Math Expert V
Joined: 02 Sep 2009
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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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1
1
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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GRE 1: Q169 V154 Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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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2
1
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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Joined: 06 Nov 2014
Posts: 1873
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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

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
Manager  S
Joined: 03 Jan 2017
Posts: 135
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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the formula says that 72 is devisible by 2^k, but not by 2^(k+1)
so, let's factor 72, 2^3 * 3^2, k is 3
Director  G
Joined: 02 Sep 2016
Posts: 649
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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The main point is what if the power of 2 in the integer 72?

Factorize 72
72=2^3*3^2

Just concerned with 2. The answer (thus) is 3.
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Intern  Joined: 25 Oct 2018
Posts: 5
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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Great job with tags here!
Intern  Joined: 16 Apr 2019
Posts: 3
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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Can someone please explain what || means? It looks like absolute value, but I though absolute value only applied when you actually put something in between the bars?
Manager  B
Joined: 25 Sep 2018
Posts: 58
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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

If k=3
72 is divisible by 2^3=8
When, 2^k+1=2^4=16 then 72/16 is not divisible.

According to condition

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Manager  B
Joined: 25 Sep 2018
Posts: 58
Re: For the positive integers a, b, and k, a^k||b means that a^k  [#permalink]

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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

Because we have another condition that 2^k+1 is not a divisor of 72.
So, if we consider option A then, it'll be 2^2+1=8 by which 72 is possible to divide.

So option A is ignored ☺

Posted from my mobile device Re: For the positive integers a, b, and k, a^k||b means that a^k   [#permalink] 10 May 2019, 12:28
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