671
658
668
645

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Well done Manish! Upvoted!

Nelson Mandela
- 5 years, 9 months ago

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Thank you for upvoting. Actually you beat me in the race to type the solution first.

Manish Dash
- 5 years, 9 months ago

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you used latex and it is presentable. My solution is messy!

Nelson Mandela
- 5 years, 9 months ago

Its a KVPY 2013 question.

(b - a) (b + a) = 2013 = 3 × 11 × 61.

ab minimum when b - a = 33 and,.

b + a = 61.

implies,a = 14 and b is 47.

Thus, minimum of ab = 14 × 47 = 658.

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Have you given KVPY ??

Ashwin Upadhyay
- 5 years, 9 months ago

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I am going to give it this year. I am in 11th class now!

Nelson Mandela
- 5 years, 9 months ago

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So how are you preparing??

Ashwin Upadhyay
- 5 years, 9 months ago

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@Ashwin Upadhyay – Its going good. how's your preparation?

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – well actually I am not preparing much for it..

Ashwin Upadhyay
- 5 years, 9 months ago

@Nelson Mandela – well i was sure about ur preparation when u told the year of paper

Ashwin Upadhyay
- 5 years, 9 months ago

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@Ashwin Upadhyay – Thanks. I actually felt the paper is good.

Nelson Mandela
- 5 years, 9 months ago

well same here

Ashwin Upadhyay
- 5 years, 9 months ago

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@Ashwin Upadhyay – good.Did you write NTSE?

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – yes but Sst was the culprit

Ashwin Upadhyay
- 5 years, 9 months ago

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@Ashwin Upadhyay – True. I got 82 in second stage!

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – are u on slack??

Ashwin Upadhyay
- 5 years, 9 months ago

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@Ashwin Upadhyay – Yes. I am on slack.

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – How much are you getting in MAT? I am also in 11th

Aditya Chauhan
- 5 years, 9 months ago

Well @Nelson Mandela , we both are classmates. In which board are you studying in ? I am also going to appear KVPY this year.

Manish Dash
- 5 years, 9 months ago

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@Manish Dash – I am from telangana state board.

What about you? @Manish Dash

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – AP State Board

Manish Dash
- 5 years, 9 months ago

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@Manish Dash – Are you from hyderabad? And which college @Manish Dash ?

Nelson Mandela
- 5 years, 9 months ago

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@Nelson Mandela – I am from Mahathi College, Vizag

Manish Dash
- 5 years, 9 months ago

@Nelson Mandela – Are you from Hyderabad?

Manish Dash
- 5 years, 9 months ago

a²-b²=2013

(a+b)(a-b)=61*33

a+b=61……………..(1)

a-b=33……………….(2)

After solving these equ the value of

a=47,b=14

ab=658

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I just factored it as a difference of squares by guessing numbers between 40 and 50.

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2013 = (b+a)(b-a)

=>(b+a) & (b-a) both are odd numbers. => One of a,b is even and the other is odd.

2013 = $3\times 11\times 61$

The following possibilities arise:

i) a+b = 61 ; b-a = 33

=> $\boxed{ a = 14 , b = 47 }$

ii) a+b = 671 ; b-a = 3

=> $\boxed{ a = 334 , b = 337 }$

iii) a+b = 61 ; b-a = 33

=> $\boxed{ a = 86 , b = 97 }$

Therefore, the minimum possible value of

abis incase (i)where bothaandbare the least. Hence the minimum possible value of $\boxed{ab = 47\times14 = 658}$