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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Prime factorization is also helpful to solve this problem. For ME it´s faster..

Probably it helps someone.

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..

Answer =B

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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Thanks Bunuel for your prompt reply.

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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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

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?

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

Let's try back solving.

A, 72 is divisible by \(2^2=4\) but 72 is also divisible by \(2^3=8\); Eliminated.

B. 72 is divisible by \(2^3=4\) but 72 is also not divisible by \(2^4=16\); Correct.

The answer is B.
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Push. Please explain the meaning of the two vertical lines || ? Never saw that in maths. It was really confusing.

Question: 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:

Note the first line - this is where they have DEFINED the meaning of '||' (it is not the same as modulus / absolute value in this context):

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

A possible example: 2^3 is a divisor of 24 but 2^(3+1) i.e. 2^4 is NOT a divisor of 24
Here: a = 2, k = 3 and b = 24

We have been given: 2^k||72
Here, a = 2, b = 72

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

We know that 72 = 8 * 9 = 2^3 * 3^2

Thus, we can say for sure that:
2^3 is a divisor of 72, but 2^(3+1) is NOT a divisor of 72
[a^k is a divisor of b, but a^(k+1) is NOT a divisor of b]

Thus, k = 3

Answer B
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Hey Bunuel ,

We know that Dividend = Divisor x Quotient + Remainder.
Now in the question, it is mentioned that 2^k is the divisor of 72, so if we write as:

72 = 2^3*3^2 = 2^k x Quotient + Remainder

So will the remainder come into the picture? Where am I going wrong?

Thanks,
Romil.
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romil666, if 2^k is a divisor of 72, that means that there is no remainder, k=3 here and there is no remainder when you divide 72 by 2^3.

Hope this helps!

Quick question here...

When I first solved this question, I thought to myself: What is the value of 2^k that gets me close to 72 with the next value being over. 2^6=64 and 2^7=128 so my inclination was that the answer should be 6.

Obviously that wasn't an answer choice. What in the question tells us to break apart 72 into its prime factors versus thinking of the highest value of 2^k independent of prime values like I did?

Dear romil666
your equation is utterly correct. Take into consideration the question stem:
a^k||b means that a^k is a divisor of b, but a^(k + 1) is not a divisor of b
In other words, b/ a^k = Integer, and the remainder is "0".
Conversely, b/a^(k + 1) is not an integer and includes some remainder.

From your equation:
72 = 2^3*3^2 = 2^k x Quotient + "0"

Hope it helps.

Could someone please help me understand this question, by rephrasing it? I haven't seen one like this before.

Many thanks!

Dear sebarm95

from question stem:
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


1. x^(y*z) signifies that x^y is a factor of "z" in other words z/x^y = Integer
2. Yet, x^(y+1) is not a factor of "z" means that x^(y+1)/z = NOT an integer

If 2^(y*72) then y is equal to what?
72/2^y = Integer then max value of y is 3 because 72 = 2^3 * 3^2.
There is only 3 twos in 72, and, thus, if y > 3, for instance 4 then 2^4 = 16 but 72/16 is not an integer. Hence, 72/2^3 is only solution in which y = 3

Hope it helps.