If r and s are positive integers such that (2^r)(4^s) = 16, then 2r +

Question

If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + s =

(A) 2 
(B) 3 
(C) 4 
(D) 5 
(E) 6

chetan2u · Accepted Answer

lets get the eq into simplest orm..
 (2^r)(4^s) = 16 ..
 (2^r)(2^{2s}) = 2^4 ..
or  r+2s=4 ..
since r and s are positive integers, only r as 2 and s as 1 satisfy the Equation..
so  2r+s=2*2+1=5 ..
D

EMPOWERgmatRichC · Answer

Hi All,

This question has a great 'brute force' element to it - you don't need to do any advanced math, but you have to be willing to 'play around' with the prompt to figure out what's possible (and what's not).

We're told that R and S are POSITIVE INTEGERS and that (2^R)(4^S) = 16. We're asked for the value of 2R + S....

Since the two variables are positive integers, that significantly restricts the possibilities. Each 'term' (2^R) and (4^S) will end up being a positive integer greater than 1 (remember, the variables are positive integers, so neither R nor S can equal 0 and neither 'term' can equal 1).

IF...
S = 2, then (2^R)(16) = 16 but we know that R CANNOT be 0, so this option is impossible. We now know that S can ONLY be 1...

When...
S = 1
(2^R)(4) = 16
2^R = 4
R = 2

Now we know that S=1 and R=2 is the only possible solution, so the answer to the question is (2)(2) + 1 = 5

Final Answer:  D

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

GMATinsight · Answer

(2^r)(4^s) = 16
2^(r+2s) = (2^4)

i.e. r+2s = 4
i.e. r=2 and s=1

i.e. 2r+s=2*2+1 = 5

Answer: option D

AverageGuy123 · Answer

(2^r ) (4^s) =16
=> (2^2)(4^1)=16

r=2
s=1
Hence 2(r)+s =2(2)+1=5
Hence D

ScottTargetTestPrep · Answer

We can re-express 4 as 2^2:

2^r * (2^2)^s = 2^4

2^r * 2^(2s) = 2^4

When we have an exponential equation in which the bases are the same, the exponents are equal. Thus we have:

2^(r + 2s) = 4

r + 2s = 4

Since r and s must be positive integers, we see that the only possible choice for r and s is r = 2 and s = 1 (notice that if s = 2, then r = 0, and if s > 2, then r < 0). Therefore, 2r + s = 2(2) + 1 = 5.

Answer: D

SeregaP · Answer

2^(r+2s)=2^4
since r,s are integers, r=2, s=1
2*2+1=5
D

guialain · Answer

Tricky and very nice question. I oversighted that s and r are positive.
was trying to solve an algebric equation to get 2r+s.
Pfff.......

Abhishek009 · Answer

Least possible value of s here will be 1 , as  4^2 = 16

Now, we have -

(2^r)(4^1) = 16

Or,  (2^r)(2^2) = 2^4

Or,  2^{ r + 2} = 2^4

So,  r + 2 = 4

Or,  r = 2

Then  2r + s = 2*2 + 1

Or,  2r + s = 5

Answer must be (D) 5

CyberStein · Answer

1. First step is to simplify what has been given.

(2^r) (4^s) = 16 <---can be simplified into this ---> (2^r)(2^2s) = 2^4

(2^r)(2^2s) = 2^4 is simpd further into ---> (2^r +2s) = 2^3
(2^r +2s) = 2^3 --->  r + 2s = 4

2. Second is to find the value of the variables, by isolating the above simplified expression

r = 4 - 2s (isolating r)
2(4-2s) + s = 
8-4s + s =
8 - 3s =
8 = 3s
8/3 = s (value of s) 
r = 4 - 2(8/3)
r= 2/3(value of r)

3. finally, plug in the values of r and s

2(2/3) + 8/3
2 2/3 + 2 2/3 = 5
 therefore the answer is D

Anantz · Answer

If there was 8 in option then it would be a problem.

I mean s=0 and r=4 making sum 8

Posted from my mobile device

hudhudaa · Answer

Nice question. I solved it the following way:

(2^r) (4^s) = 16
(2^r) (4^s) = 2^2 * 4^1

therefore, this means:
r = 2 and s = 1

plug into the target question:
2r + s = ?
2(2) + 1 = 5

EMPOWERgmatRichC · Answer

Hi Anantz,

If this question were to appear on Test Day, then it's certainly possible that '8' could be among the answer choices. However, we're told that S and R are POSITIVE INTEGERS, so S=0, R=4 is NOT an option here (and '8' is not a correct answer). In that same way, S=2, R=0 is not a correct answer either (even though the answer "2" does appear among the five answer choices).

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

AliciaSierra · Answer

Bunuel , as per Wiley Efficient Learning (app.efficientlearning.com), it is sub-600 level question.

Radheya · Answer

=> (2^r)(4^s) = 16
As, 4^s = (2)^2s 
=> (2^r)(2^2s) = (2^2)(2^2)

So, r = 2 and 2s = 2, so s = 1.

2r + s = 2(2) + 1 = 4 + 1
= 5

Option D.

Kinshook · Answer

Given: r and s are positive integers such that (2^r)(4^s) = 16

Asked: 2r + s = ?

2^{r+2s} = 16 = 2^4 
r+2s = 4
r =2, s=1

2r + s = 2*2 + 1 = 5

IMO D

gagansh1840 · Answer

only possible value of r and s are 2 and 1. hence 2r+s=5

MHIKER · Answer

r+2s=4  
2+2*1=4

So, 2r+s=2*2+1=5

Ans. D

shardsingh1134527 · Answer

2^r*4^s= 16= 2^2 * 4^1 
r=2, s=1
Now we can solve easily.

Paras96 · Answer

(2^r)(4^s) = 16
=>(2^r)(2^2s) = 2^4
=> r +2s = 4 (Since r & s are positive integers)
=> r = 2 and s = 1

Therefore 2r+s=5

Hence D

If r and s are positive integers such that (2^r)(4^s) = 16, then 2r +

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