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If a, b are natural numbers such that 2013 + a 2 = b 2 , then the minimum possible value of ab is Sol. (b - a) (b + a) = 2013 = 3 × 11 × 61 ( In what logic in their mind they split this ) ab minimum when b - a = 33 ( I cant understand this step please explain) b + a = 61 a = 14

  1. If a, b are natural numbers such that 2013 + a2 = b2, then the minimum possible value of ab is
Sol. (b - a) (b + a) = 2013 = 3 × 11 × 61 (In what logic in their mind they split this )
ab minimum when b - a = 33 ( I cant understand this step  please explain)
b + a = 61
a = 14

Grade:11

2 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
Given a2+2013=b2
it can also be written as b2-a2=2013
by prime factorization we have
b2-a2=3*11*61
so (b+a)(b-a)=3*11*61
as they asked minimum possible value of ab
product is minimum when the difference between the numbers is less
as the difference increases the value of their product also increases
So least product from 3 numbers 3,11,61 is 33
So b-a=33 and b+a=61
SO we get a=14 and b=47

vikas
16 Points
3 years ago
hi,
given a^2+2013=b^2
b^2-a^2=2013
by prime factorisation,
b^2-a^2=3*11*61
(b-a)(b+a)=(33)(61)
on comparing,
b-a=33 and b+a=67
             b-a=33  +
            b+a=61
2b=94
b=47
a=61-47=14    [since a=61-b]
hence,
ab=14*47=658

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