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

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Math Expert
Joined: 02 Sep 2009
Posts: 53063
If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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16 Feb 2015, 05:56
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25% (medium)

Question Stats:

77% (01:39) correct 23% (02:17) wrong based on 115 sessions

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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.

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Joined: 10 Jan 2015
Posts: 4
Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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

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16 Feb 2015, 07:07
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(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)

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Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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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
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Re: If a and b are positive integers, is √(a+b) an integer?  [#permalink]

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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.

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Math Expert
Joined: 02 Sep 2009
Posts: 53063
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.
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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?   [#permalink] 17 Dec 2017, 14:09
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# If a and b are positive integers, is √(a+b) an integer?

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