2013! Sorry I forgot what is the present year
If a , b are natural numbers such that 2 0 1 3 = b 2 − a 2 , then the minimum possible value of a b is:
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4 solutions
Well done Manish! Upvoted!
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Thank you for upvoting. Actually you beat me in the race to type the solution first.
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you used latex and it is presentable. My solution is messy!
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.
Have you given KVPY ??
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I am going to give it this year. I am in 11th class now!
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So how are you preparing??
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@Ashwin Upadhyay – Its going good. how's your preparation?
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@Nelson Mandela – well actually I am not preparing much for it..
@Nelson Mandela – well i was sure about ur preparation when u told the year of paper
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@Ashwin Upadhyay – Thanks. I actually felt the paper is good.
well same here
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@Ashwin Upadhyay – good.Did you write NTSE?
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@Nelson Mandela – yes but Sst was the culprit
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@Ashwin Upadhyay – True. I got 82 in second stage!
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@Nelson Mandela – are u on slack??
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@Ashwin Upadhyay – Yes. I am on slack.
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@Nelson Mandela – How much are you getting in MAT? I am also in 11th
Well @Nelson Mandela , we both are classmates. In which board are you studying in ? I am also going to appear KVPY this year.
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@Manish Dash – I am from telangana state board.
What about you? @Manish Dash
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@Nelson Mandela – AP State Board
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@Manish Dash – Are you from hyderabad? And which college @Manish Dash ?
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@Nelson Mandela – I am from Mahathi College, Vizag
@Nelson Mandela – Are you from Hyderabad?
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
I just factored it as a difference of squares by guessing numbers between 40 and 50.
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 × 1 1 × 6 1
The following possibilities arise:
i) a+b = 61 ; b-a = 33
=> a = 1 4 , b = 4 7
ii) a+b = 671 ; b-a = 3
=> a = 3 3 4 , b = 3 3 7
iii) a+b = 61 ; b-a = 33
=> a = 8 6 , b = 9 7
Therefore, the minimum possible value of ab is in case (i) where both a and b are the least. Hence the minimum possible value of a b = 4 7 × 1 4 = 6 5 8