Question

There is a natural number which becomes equal to the square of a natural number when $100$ is added to it, and to the square of another natural number when $168$ is added to it. Find the number.

Answer 1

From the question, we can write that

$a$ + $100$ = $b^2$
$a$ + $168$ = $c^2$
=> $c^2-b^2$ = 68
=> $(c-b)(c+b)=1\times 68$

Now solve the equation for $(c-b)=1$ and $(c+b)=68$

We get $c=35$ and $b=34$. (here $c=34.5$ which is $35$ and $b=33.5$ which can be written as $34$ by ignoring the point )

$a+100=b^2$

$a={34}^2-100$

$a=1156-100=1056$ hence this the required number.