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# If A and B are integers and A^2-B^2=101, what is A?

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Math Expert
Joined: 02 Aug 2009
Posts: 7757
If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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13 Apr 2016, 03:04
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95% (hard)

Question Stats:

13% (02:08) correct 87% (01:45) wrong based on 60 sessions

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If A and B are integers and $$A^2-B^2=101$$, what is A?
(1) A and B are consecutive integers.
(2) A>B

OA in 2 days

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Re: If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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13 Apr 2016, 03:29
1
(A + B)(A - B) = 101; A = ?
101 is a prime number and hence 101 can be factorized in two ways --> (101 * 1) or (-101 * -1)
Case 1: A + B = 101
A - B = 1
A = 51; B = 50

Case 2: A + B = -101
A - B = -1
A = -51; B = -50

Case 3: A + B = 1
A - B = 101
A = 51; B = -50

Case 4: A + B = -1
A - B = -101
A = -51; B = 50

St1: A and B are consecutive. --> Not Sufficient as A can still have two values A = -51 or 51

St2: A > B --> A can take only one value = 51 --> Sufficient

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Posts: 7757
Re: If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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15 Apr 2016, 09:12
1
Vyshak wrote:
(A + B)(A - B) = 101; A = ?
101 is a prime number and hence 101 can be factorized in two ways --> (101 * 1) or (-101 * -1)
A + B = 101
A - B = 1
A = 51; B = 50

A + B = -101
A - B = -1
A = -51; B = -50

St1: A and B are consecutive. --> Not Sufficient as A can still have two values A = -51 or 51

St2: A > B --> A can take only one value = 51 --> Sufficient

Hi Vyshak,
your answer is correct But there are two more case which you have missed out..
1) A = -51 and B=50..
A+B = -1 and A-B=-101..

2) A= 51 and B =-50
A+B = 1 and A-B=101..

say Q was what is the value of B..
with the cases you have taken , you would have got the answer as B with B's value as 50, but that would have been wrong
the two cases where A >Bwould be
A=51 B=50 OR A=51 B=-50
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Re: If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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15 Apr 2016, 09:41
chetan2u wrote:
Vyshak wrote:
(A + B)(A - B) = 101; A = ?
101 is a prime number and hence 101 can be factorized in two ways --> (101 * 1) or (-101 * -1)
A + B = 101
A - B = 1
A = 51; B = 50

A + B = -101
A - B = -1
A = -51; B = -50

St1: A and B are consecutive. --> Not Sufficient as A can still have two values A = -51 or 51

St2: A > B --> A can take only one value = 51 --> Sufficient

Hi Vyshak,
your answer is correct But there are two more case which you have missed out..
1) A = -51 and B=50..
A+B = -1 and A-B=-101..

2) A= 51 and B =-50
A+B = 1 and A-B=101..

say Q was what is the value of B..
with the cases you have taken , you would have got the answer as B with B's value as 50, but that would have been wrong
the two cases where A >Bwould be
A=51 B=50 OR A=51 B=-50

Thanks chetan2u. Added the two more cases. As you have suggested above, I would have got this answer wrong if the question asked for the value of B as I failed to notice 2 more cases while answering the question. Liked this question!!
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Re: If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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16 Apr 2016, 02:41
If A and B are integers and A 2 −B 2 =101 A2−B2=101 , what is A?
(1) A and B are consecutive integers.
(2) A>B

There are 2 variables in the original condition (A and B) and 1 equation (A^2-B^2=101) in the original condition. In order to match the number of variables to the number of equations, we need 1 more equation. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that D is the answer.
In case of the condition 1), from (A-B)(A+B)=101, we get A+B=101 and A-B=1. Then we get A=51 and B=50. However, when A+B=-101 and A-B=-1, we get A=-51 and B=-50. The answers are not unique and the conditions are not sufficient.
In case of the condition 2), we get (A,B)=(51,50),(51,-50). However, since A=51, the answer is unique and the condition is sufficient. Therefore, the correct answer is B.

l For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If A and B are integers and A^2-B^2=101, what is A?  [#permalink]

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25 Dec 2018, 09:05
chetan2u wrote:
If A and B are integers and $$A^2-B^2=101$$, what is A?
(1) A and B are consecutive integers.
(2) A>B

OA in 2 days

Dear Moderator,
Can we have the OA for this question. Thank you.
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Re: If A and B are integers and A^2-B^2=101, what is A?   [#permalink] 25 Dec 2018, 09:05
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