**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,

x

**+ (x + 1)**^{2}**= 113**^{2}
∴ x

**+ x**^{2 }**+ 2x + 1 = 113**^{2}
∴ 2x

**+ 2x + 1 – 113 = 0**^{2}
∴ 2x

**+ 2x – 112 = 0**^{2}
Dividing by 2 we get,

x

**+ x – 56 = 0**^{2}
∴ x

**+ 8x – 7x – 56 = 0**^{2}
∴ 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