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For the positive integers a, b (OG16)

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by OptimusPrep » Thu May 19, 2016 7:58 pm
boomgoesthegmat 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

OA: B
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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by ceilidh.erickson » Mon Oct 09, 2017 7:57 am
Any positive integer can be expressed as the product of prime factors - we call this PRIME FACTORIZATION. For example, the prime factorization of 140 is (2^2)(5^1)(7^1), and 108 = (2^2)(3^3). This prime factorization is really helpful when you're solving questions with variable exponents. For example:

12 = (2^x)(3^y)
In order to solve for x and y here, we need to break 12 down to its prime factors:
12 = (2^2)(3)
So, (2^2)(3) = (2^x)(3^y)
Therefore x = 2 and y = 1.

In #110, we have "k" and "k + 1" as exponents, which is a big clue that we probably want to think in terms of prime factorization.

The hardest part about this question is simply figuring out what it's asking - breaking down "a^k is a divisor of b, but a^(k + 1) is not." How do we translate that? So "a" to a certain exponent goes evenly into "b," but "a" to the next highest exponent does not. That would mean that a^k was the maximum number of a's that go into "b." In other words... all the a's!

So, 2^k || 72 would be the maximum number of factors of 2 that go into 72 - all the 2's in 72. If we factor 72, we find that the prime factorization is (2^3)(3^2). There are 3 factors of 2 in 72 (that's what 2^3 tells us), so k must equal 3.

There's more on prime factorization here: https://www.beatthegmat.com/2-x-2-x-2-3- ... tml#578272
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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by [email protected] » Sun Mar 18, 2018 7:18 pm
Hi All,

We're told that A, B and K are all POSITIVE INTEGERS and A^K || B means that A^K is a divisor of B, but A^(K+1) is NOT a divisor of B. Using the defined symbol - and the fact that 2^k ||72 - we're asked to find the value of K. While this question might 'look scary', it can be solved rather easily by TESTing THE ANSWERS. We'll start with the 'easiest' answer and work our way through the rest until we find the one that 'fits' everything we're told...

Answer A: 2
For 2 to be the correct answer, (2^2) MUST be a factor of 72 and (2^3) must NOT.
72/4 = 18
72/8 = 9
Both numbers ARE factors, so this CANNOT be the answer.

Answer A: 3
For 3 to be the correct answer, (3^2) MUST be a factor of 72 and (3^3) must NOT.
72/9 = 8
72/27 = 2 r 18
The first number IS a factor and the second number is NOT, so this IS the answer.

Final Answer: B

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by Jeff@TargetTestPrep » Tue Mar 20, 2018 4:07 pm
boomgoesthegmat 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
We see that 2^k is a divisor of 72, but 2^(k+1) is not.

Since 72 = 3^2 x 2^3, k must be 3 since 2^3 is a divisor of 72, but 2^4 is not.

Answer: B

Jeffrey Miller
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[email protected]

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by deloitte247 » Sun Apr 01, 2018 9:50 am
This requires a trial and error approach.
For any of the options given, the answer is the option that can divide 72, and the option plus one cannot divide 72, when raised to 2.

option A: 2
2^2=4, 4 can divide 72 to give 18.
2^(2+1)=2^3=8, 8 can also divide 72 to give 9.
Therefore, option A wrong

option B: 3
2^3=8, 8 can divide 72 to give 9
2^(3+1)=2^4=16, 16 cannot divide 72
Therefore, option B is correct

option C: 4
2^4=16, 16 cannot divide 72
Therefore, option C is wrong since the first condition is incorrect is incorrect

option D and E are similarly incorrect.

Therefore, answer=3 (option B)
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Re: For the positive integers a, b (OG16)

by iammohakjain » Wed Jul 01, 2020 9:37 am
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
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Re: For the positive integers a, b (OG16)

by Scott@TargetTestPrep » Sat Jul 04, 2020 8:56 am
boomgoesthegmat wrote: ↑
Thu May 19, 2016 3:43 pm
 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

OA: B
Solution:

We see that 2^k is a divisor of 72, but 2^(k+1) is not.

Since 72 = 3^2 x 2^3, k must be 3 since 2^3 is a divisor of 72, but 2^4 is not.

Answer: B

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