KVPY SA Exam 2013 Question Paper

Question : 61 of 80

Marks: +1, -0

PART II

MATHEMATICS

Let a ,b ,c ,d ,e be natural numbers in an arithmetic progression such that a + b + c + d + e is the cube of an integer and b + c + d is square of an integer. The least possible value of the number of digits of c is

Validate

Solution:

a = C - 2D
b = C - D
c = C
d = C + D
e = C + 2D
a + b + c + d + e = 5c =

λ^3

3 b + c + d = 3c =

µ^2

⇒3λ^3==5µ^2

λ^3/5=µ^2/3

least possibility
λ = 5 × 3 = 15, µ = 5 × 3 × 3 = 45

\C=45^2/3=675

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