What is the greatest 6 digit number that is a perfect square?

Introduction / Context:
This question involves perfect squares and place value. You are asked to find the largest 6 digit number that is a perfect square. Understanding how squares grow and how to bound them by digit length is an important skill that connects number sense with algebra and estimation.

Given Data / Assumptions:
- We are interested in 6 digit numbers, which range from 100000 to 999999 inclusive.
- We want the greatest such number that is a perfect square of some integer n.

- This means n^2 should have exactly 6 digits and be as large as possible.

Concept / Approach:
To find the largest 6 digit perfect square, we look for the largest integer n such that n^2 is less than 1000000 (since 1000000 is a 7 digit number). The next integer after n must produce a square that has 7 digits. Therefore, n should be one less than the first integer whose square is at least 1000000. Since 1000^2 = 1000000, we consider n = 999 and compute 999^2. That square will be the largest 6 digit perfect square.

Step-by-Step Solution:
Step 1: Note that 1000^2 = 1000000, which is a 7 digit number.Step 2: Therefore, any integer n greater than or equal to 1000 will give n^2 with at least 7 digits.Step 3: We need the largest integer n whose square still has 6 digits, so n must be 999.Step 4: Compute 999^2. Use the identity (1000 - 1)^2 = 1000^2 - 2 * 1000 * 1 + 1.Step 5: Calculate (1000 - 1)^2 = 1000000 - 2000 + 1.Step 6: Simplify: 1000000 - 2000 = 998000, then 998000 + 1 = 998001.Step 7: So 999^2 = 998001, which is a 6 digit number.Step 8: Check that it is the greatest 6 digit square by confirming that the next integer 1000 has square 1000000, which is 7 digits and therefore too large.

Verification / Alternative check:
You can compare 998001 with other options. For example, 999001 is close but would require the square root to be slightly more than 999, and 1000^2 already jumps to 1000000, so 999001 cannot be a perfect square. Similarly, 998009 and 998101 are not equal to 999^2 and would not have integer square roots. The calculation based on the binomial square formula is precise and confirms that 998001 is indeed the square of 999.

Why Other Options Are Wrong:
Option 999001 is not equal to 999^2, and there is no integer whose square produces 999001 within the 6 digit range. Options 998009 and 998101 are also not perfect squares of any integer (they lie between consecutive square values). The value 997969 is lower than 998001 and may be a square of a slightly smaller integer, but it is not the greatest 6 digit square. Only 998001 corresponds to 999^2, which is the largest 6 digit perfect square.

Common Pitfalls:
Some learners guess among the choices without connecting them to specific square roots, or they attempt to take square roots by trial and error without estimation. Others may incorrectly think 999999 is a square, but the square root of 999999 is not an integer. Understanding that 1000^2 is the first 7 digit square and stepping down to 999 is the systematic way to reach the correct answer.

Final Answer:
The greatest 6 digit perfect square number is 998001.