Question

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

Answer 1

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

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

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

$\Rightarrow 2x^2+2x-312=0$

$\Rightarrow x^2+x-156=0$

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

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

$\Rightarrow x+13=0$ or $x-12=0$

$\Rightarrow x=-13$ or $12$

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

$\therefore x=12$

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