The sum of the squares of two consecutive natural numbers is 113. Find the numbers.

1. The sum of the squares of two consecutive natural numbers is 113. Find the numbers.

Sol. Let the two consecutive natural numbers be x and x + 1

As per the given condition,

x2 + (x + 1)2 = 113

 x2 + x2 + 2x + 1 = 113

2x2 + 2x + 1 – 113 = 0

2x2 + 2x – 112 = 0

Dividing by 2 we get,

x2 + x – 56 = 0

x2 + 8x – 7x – 56 = 0

x (x + 8) – 7 (x + 8) = 0

(x + 8) (x – 7) = 0

x + 8 = 0 or x – 7 = 0

x = – 8 or x = 7

∵  x is a natural number

∴ x cannot be negative

∴  x = 7

And x + 1 = 7 + 1 = 8


The two consecutive natural numbers are 7 and 8 
respectively