GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Aug 2018, 21:52

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If a and b are positive integers, is √(a+b) an integer?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 47977
If a and b are positive integers, is √(a+b) an integer?  [#permalink]

### Show Tags

16 Feb 2015, 06:56
00:00

Difficulty:

25% (medium)

Question Stats:

75% (01:15) correct 25% (01:49) wrong based on 110 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, 08:06
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, 08:07
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)

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, 11:58
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

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12189
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, 18:36
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.

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
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Math Expert
Joined: 02 Sep 2009
Posts: 47977
Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

### Show Tags

22 Feb 2015, 11:20
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: 367
Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

### Show Tags

17 Dec 2017, 15:09
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, 15:09
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.