The Problem

A general wants to draw up his 36562 soldiers in the form of a solid square. However, after arranging them, he finds that some soldiers are left over. The task is to determine how many soldiers are left over and provide a detailed explanation.

Solution

To solve this problem, we need to find the number of soldiers left after arranging them in a solid square. Let's break down the solution into smaller steps:

Step 1: Finding the largest square possible
To arrange the soldiers in the form of a solid square, we need to find the largest square possible. We can do this by finding the square root of the total number of soldiers.

√36562 ≈ 191.179

So, the largest square possible will have sides of length 191 soldiers.

Step 2: Calculating the number of soldiers in the square
To find the number of soldiers in the square, we need to calculate the area of the square. The area of a square is given by the formula:

Area = side length × side length

Area = 191 × 191 = 36481

Therefore, there are 36481 soldiers in the square.

Step 3: Finding the number of soldiers left
To determine the number of soldiers left, we subtract the number of soldiers in the square from the total number of soldiers.

Number of soldiers left = Total number of soldiers - Number of soldiers in the square

Number of soldiers left = 36562 - 36481 = 81

Hence, there are 81 soldiers left after arranging them in a solid square.

Conclusion
After arranging the soldiers in the form of a solid square, the general found that 81 soldiers were left over. This was determined by finding the largest square possible and calculating the number of soldiers in the square. By subtracting this number from the total number of soldiers, we obtained the number of soldiers left.