The greatest $5$ -digit number which is a perfect square is:

The greatest $5$ -digit number is $99999$. Let's find its square root using the long division method to find out how much it exceeds the nearest lower perfect square:
Taking the square root of $99999$:

* $316^2=99856$
* $317^2=100489$ (which is a $6$ -digit number)

Let's check the subtraction:

$$99999-99856=143$$

Thus, the greatest $5$ -digit perfect square is $316^2=99856$.