Question

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

Answer 1

Let the two consecutive natural numbers be x and (x + 1). Then, by the given condition,
x² + (x + 1)² = 113
$\Rightarrow$ x² + x² +2x + 1 = 113
$\Rightarrow$ 2x² + 2x – 112 = 0
$\Rightarrow$ 2(x² + x – 56) = 0
$\Rightarrow$ x² + x – 56 = 0
On splitting the middle term x as 8x – 7x, we get:
x² + 8x – 7x – 56 = 0
$\Rightarrow$ x(x + 8) – 7(x + 8) = 0
$\Rightarrow$ (x + 8)(x – 7) = 0
$\Rightarrow$ x + 8 = 0 or x – 7 = 0
$\Rightarrow$ x = –8 or x = 7
Since x is a natural number, it cannot be negative.
Therefore, x = 7 and (x + 1) = 8
Thus, the two consecutive natural numbers are 7 and 8.