A general of an army wants to form his 36562 soldiers into the largest possible perfect square formation. After arranging them in this square, some army men remain unused. How many soldiers remain unused outside the square formation?

Introduction / Context:
This arithmetic reasoning question deals with square numbers and remainders. The general has 36562 soldiers and wants to arrange them in a perfect square formation, which requires the same number of soldiers in each row and each column. However, the total number of soldiers is not itself a perfect square, so some soldiers will remain unused. The task is to find how many soldiers are left over after forming the largest possible square.

Given Data / Assumptions:
- Total number of soldiers: 36562. - Soldiers are to be arranged in a perfect square grid. - Each side of the square must contain the same integer number of soldiers. - The general wants to use as many soldiers as possible in the square. - We must find the number of soldiers who remain outside this largest possible square.

Concept / Approach:
To solve this, we find the largest perfect square less than or equal to 36562. If k^2 is this largest square, then k soldiers stand in each row and k soldiers stand in each column, using k^2 soldiers in total. The number of unused soldiers is then the difference 36562 - k^2. Therefore, we need to approximate the square root of 36562, find the nearest integer k such that k^2 is less than or equal to 36562, and compute the remainder.

Step-by-Step Solution:
Step 1: Estimate the square root of 36562. We know 190^2 = 36100 and 200^2 = 40000, so the square root lies between 190 and 200. Step 2: Compute squares around this range more precisely. Calculate 191^2 and 192^2. Step 3: 191^2 = 191 * 191 = 36481. Step 4: 192^2 = 192 * 192 = 36864, which is greater than 36562. Step 5: Therefore, 191^2 is the largest perfect square less than 36562. Step 6: Find the remainder: 36562 - 36481 = 81. These 81 soldiers cannot fit into the square and remain unused.

Verification / Alternative check:
We can verify the calculations. First, recompute 191^2 using (190 + 1)^2 = 190^2 + 2 * 190 * 1 + 1^2. This gives 190^2 = 36100, 2 * 190 = 380 and 1^2 = 1, so 36100 + 380 + 1 = 36481, confirming 191^2. Next, verify that 192^2 is larger than the total number of soldiers; with 36864 it clearly exceeds 36562, so any larger k would also be invalid. Finally, verify the subtraction 36562 - 36481. Subtracting 36481 from 36562 yields 81, confirming our remainder.

Why Other Options Are Wrong:
- 36, 65 and 72 are all smaller than the actual remainder and do not satisfy 36562 - k^2 for any integer k near the square root. - 97 is larger than the true remainder and would correspond to a smaller square than 191^2, which contradicts the requirement for the largest possible square formation. Only 81 matches the difference between the total soldiers and the largest perfect square not exceeding that total.

Common Pitfalls:
Many students make mistakes in squaring large numbers or estimating the square root too roughly. Some may use 190^2 = 36100 and forget to check 191^2 or 192^2, leading to an incorrect remainder. Others subtract in the wrong order and get negative or inconsistent results. Using algebraic identities for squaring and performing subtraction carefully help avoid such errors.

Final Answer:
After forming the largest possible square with 36562 soldiers, the number of soldiers remaining unused is 81.