For the positive integers a, b (OG16)

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

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

GMAT assassins aren't born, they're made,
Rich

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