The greatest five-digit number that is a perfect square can be found by determining the square root of 99999, the largest five-digit number, and rounding down to the nearest integer. Then, we square that integer to get the largest five-digit square.

1. $\sqrt{99999} \approx 316.22$

2. The largest integer less than or equal to 316.22 is 316.

3. Now, squaring 316 gives: $316^2 = 99856$

Thus, the greatest five-digit number that is a perfect square is 99856.

Would you like further explanation, or do you have any other questions?

Here are five related questions to deepen your understanding:

1. What is the smallest five-digit number that is a perfect square?
2. How can we find the largest six-digit number that is a perfect square?
3. What are some properties of perfect squares in relation to their digit length?
4. How would the process change if we wanted the largest three-digit or four-digit square?
5. Can there be any pattern observed among the largest n-digit perfect squares?

Tip: To quickly check if a number is a perfect square, you can calculate its square root and see if it is a whole number.