Question

The sum of the squares of two consecutive natural numbers is $313$. Find the numbers.

Answer 1

Let $x$ and $x+1$ be the two consecutive numbers.

$x^2+{(x+1)}^2=313$

$\Rightarrow 2x^2+2x+1=313\Rightarrow x^2+x-156=0\Rightarrow x^2+13x-12x-156=0$

$\Rightarrow x(x+13)-12(x+13)=0\Rightarrow x=12orx=-13$

Since, $x$ being a natural number, cannot be a negative number. Therefore $x=12$

Hence, the two consecutive natural numbers are $12$ and $13$