Determine the largest four digit number that is a perfect square.

Introduction / Context:
This question checks your understanding of perfect squares and the range of four digit numbers. You are asked to identify the largest four digit number that can be written as the square of an integer. This is a standard type of problem that appears in many quantitative aptitude exams to test number sense and familiarity with square values near a boundary like 10000.

Given Data / Assumptions:

• We are concerned only with four digit numbers, that is numbers from 1000 to 9999.
• We want the largest number in this range that is a perfect square.
• Numbers greater than 9999 are five digit numbers and are not allowed.
• We consider positive integer squares only.

Concept / Approach:
The smallest five digit number is 10000, which is 100^2. Therefore, the largest four digit perfect square must be the square of 99, since the next integer 100 already yields a five digit result. So the problem reduces to calculating 99^2 accurately. Once we have that value, we simply choose it from among the given options.

Step-by-Step Solution:
Step 1: Note that 100^2 = 10000, which is a five digit number. Step 2: Therefore, the square of any integer greater than or equal to 100 will be at least 10000 and hence not a four digit number. Step 3: The largest integer whose square could still be four digits is 99. Step 4: Compute 99^2. One way is to use the identity (100 − 1)^2 = 100^2 − 2 * 100 + 1. Step 5: Substitute 100: (100 − 1)^2 = 10000 − 200 + 1 = 9801. Step 6: Therefore 99^2 = 9801. Step 7: So 9801 is the largest possible four digit perfect square.

Verification / Alternative check:
You can confirm this by checking the next perfect square after 9801, which is 100^2 = 10000. Since 10000 is outside the four digit range, 9801 is indeed the maximum within that range. You can also verify that 9801 is a square by checking that 99 × 99 equals 9801 using standard multiplication. No larger number under 10000 will be a square, because that would require an integer larger than 99 whose square is still four digits, which is impossible.

Why Other Options Are Wrong:
The number 9999 is not a perfect square because there is no integer n such that n^2 = 9999. Similarly, 9702, 9604, and 9409 are not squares of integers close to 99 or 100. In fact, 98^2 = 9604 and 97^2 = 9409. While these two are perfect squares, they are smaller than 9801, so they cannot be the largest four digit perfect square. Among all four digit squares, 9801 stands above the rest.

Common Pitfalls:
A common mistake is to assume 9999 is a perfect square because it is the largest four digit number. Without checking, some students may choose it by intuition. Others might compute 98^2 or 97^2 and forget to check 99^2 using the identity for (100 − 1)^2. Another possible error is incorrect multiplication when computing 99^2 manually, leading to values like 9901 or 9701. Using the algebraic identity or carefully performing multiplication avoids these issues.

Final Answer:
The largest four digit number that is a perfect square is 9801.