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• $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) If a and b are positive integers, is √(a+b) an integer?  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 50578 If a and b are positive integers, is √(a+b) an integer? [#permalink] Show Tags 16 Feb 2015, 05:56 00:00 Difficulty: 25% (medium) Question Stats: 76% (01:38) correct 24% (02:17) wrong based on 113 sessions HideShow timer Statistics If a and b are positive integers, is √(a+b) an integer? (1) a < b + 3 (2) a = b(b − 1) Kudos for a correct solution. _________________ Intern Joined: 10 Jan 2015 Posts: 4 Re: If a and b are positive integers, is √(a+b) an integer? [#permalink] Show Tags 16 Feb 2015, 07:06 1 Bunuel wrote: If a and b are positive integers, is √(a+b) an integer? (1) a < b + 3 (2) a = b(b − 1) Kudos for a correct solution. Ans. B A) say b=1then possible values of a are 1,2,3 and we don't have a definite ans. b)The expression reduces to a+b=b^2, and as a and b are +ve integers , so B sufficient. Ans B Manager Joined: 14 Sep 2014 Posts: 105 Concentration: Technology, Finance WE: Analyst (Other) If a and b are positive integers, is √(a+b) an integer? [#permalink] Show Tags 16 Feb 2015, 07:07 1 (1) a < b + 3 [Insufficient] a and b could both be 2 and $$\sqrt{(2+2)}$$ is the integer 2. a and b could both be 3 and $$\sqrt{3+3}$$ is $$\sqrt{6}$$, which is not an integer. (2) a = b(b − 1) [Sufficient] $$a = (b^2 - b)$$ so $$(a + b) = (b^2 - b + b) = b^2$$. $$\sqrt{b^2} = b$$, which is an integer (as stated in the problem) The correct answer is B. Intern Joined: 18 Jan 2015 Posts: 4 Location: United States (MN) Concentration: General Management, Entrepreneurship Schools: Carlson GMAT Date: 04-27-2015 GPA: 3.76 WE: Engineering (Manufacturing) Re: If a and b are positive integers, is √(a+b) an integer? [#permalink] Show Tags 16 Feb 2015, 10:58 1 If √(a+b) is an integer then, (a+b) has to be a perfect square of a number, lets say x. => (a+b) = x^2 Now, (1) says a< b+3, so substitute a=b+2 in the above eqn. (b+2+b) = x^2 which means 2(b+1) = x^2, this still doesn't tell me anything... so not sufficient. (2)substitute a =b(b-1) in main eqn. (b^2 -b +b) = x^2 which means b^2 = x^2. because RHS is an int. so is LHS. thus sufficient ans is B _________________ Two Years at Harvard Business School Ahead of the Curve EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12853 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If a and b are positive integers, is √(a+b) an integer? [#permalink] Show Tags 16 Feb 2015, 17:36 1 Hi All, This question can be solved by TESTing VALUES and pattern-matching. We're told that A and B are POSITIVE INTEGERS. We're asked if \sqrt{(A+B} is an INTEGER. This is a YES/NO question. Fact 1: A < B+3 IF..... A = 1 B = 3 Then \sqrt{(1+3)} = 2 and the answer to the question is YES. IF.... A = 1 B = 1 Then \sqrt{(1+1)} = NOT an integer and the answer to the question is NO. Fact 1 is INSUFFICIENT Fact: 2: A = B(B-1) IF.... B = 2 A = 2(1) = 2 Then \sqrt{(2+2)} = 2 and the answer to the question is YES. IF.... B = 3 A = (3)(2) = 6 Then \sqrt{(6+3)} = 3 and the answer to the question is YES. IF.... B = 4 A = (4)(3) = 12 Then \sqrt{(12+4)} = 4 and the answer to the question is YES. As you can see, as B increases, we ALWAYS end up with a perfect square "under" the square root, so we ALWAYS end up with an integer in the end. Fact 2 is SUFFICIENT. Final Answer: 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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Math Expert
Joined: 02 Sep 2009
Posts: 50578
Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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22 Feb 2015, 10:20
Bunuel wrote:
If a and b are positive integers, is √(a+b) an integer?

(1) a < b + 3
(2) a = b(b − 1)

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

I. If we pick numbers for a and b it is possible to create a scenario where $$\sqrt{(a+b)}$$ (a = 1 and b = 3) and a scenario where $$\sqrt{(a+b)}$$ is not an integer (a = 1 and b = 4). Not Sufficient.

II. If we expand a = b(b-1), we get a = b²-b. Then we can substitute b²-b for a in the original expression and we get √(b²) - b + b or √(b²) which is equal to b. Since b is an integer, $$\sqrt{(a+b)}$$ is an integer. Sufficient.
_________________
Senior Manager
Joined: 15 Jan 2017
Posts: 360
Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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17 Dec 2017, 14:09
Using algebra:

1) a < b +3 --> a - b< 3 -->Or a =b, a< b. Not suff because we don't know nature of numbers

2) a = b ^2 -b
_/a + b --> _/b sq + b - b = _/b^2 = POSITIVE b as mentioned in stem

Kudos, if you like this solution
Re: If a and b are positive integers, is √(a+b) an integer? &nbs [#permalink] 17 Dec 2017, 14:09
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