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

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Math Expert
Joined: 02 Sep 2009
Posts: 44659
If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink]

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10 Jan 2016, 07:47
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45% (medium)

Question Stats:

69% (01:10) correct 31% (01:07) wrong based on 746 sessions

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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
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Aug 2009
Posts: 5777
Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink]

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10 Jan 2016, 08:11
5
KUDOS
Expert's post
Bunuel wrote:
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

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
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink]

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13 Jan 2016, 16:41
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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

[Reveal] Spoiler:
D

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** SVP Joined: 08 Jul 2010 Posts: 2068 Location: India GMAT: INSIGHT WE: Education (Education) If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 13 Jan 2016, 23:33 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED Bunuel wrote: 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 (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 _________________ Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html 22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION Manager Joined: 24 Jun 2014 Posts: 52 Concentration: Social Entrepreneurship, Nonprofit Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 15 Jan 2016, 02:01 (2^r ) (4^s) =16 => (2^2)(4^1)=16 r=2 s=1 Hence 2(r)+s =2(2)+1=5 Hence D Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 2480 Location: United States (CA) Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 15 Mar 2017, 16:34 1 This post received KUDOS Expert's post Bunuel wrote: 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 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 _________________ Scott Woodbury-Stewart Founder and CEO GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Manager Joined: 03 Jan 2017 Posts: 184 Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 18 Mar 2017, 14:58 2^(r+2s)=2^4 since r,s are integers, r=2, s=1 2*2+1=5 D Manager Joined: 01 Dec 2016 Posts: 121 Concentration: Finance, Entrepreneurship GMAT 1: 650 Q47 V34 WE: Investment Banking (Investment Banking) Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 22 Mar 2017, 00:10 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....... _________________ What was previously considered impossible is now obvious reality. In the past, people used to open doors with their hands. Today, doors open "by magic" when people approach them Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 3405 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 22 Mar 2017, 08:43 1 This post was BOOKMARKED Bunuel wrote: 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 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 _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only ) Manager Joined: 21 Jun 2017 Posts: 78 Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 07 Oct 2017, 02:49 Bunuel wrote: 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 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 Intern Joined: 16 Sep 2017 Posts: 10 Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 07 Nov 2017, 06:46 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 Intern Joined: 27 Apr 2016 Posts: 6 Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 19 Feb 2018, 10:52 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 EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 11515 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r + [#permalink] ### Show Tags 21 Feb 2018, 20:01 Anantz wrote: 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 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 _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: If r and s are positive integers such that (2^r)(4^s) = 16, then 2r +   [#permalink] 21 Feb 2018, 20:01
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