A General wishes to draw up his $36581$ soldiers in the form of a solid square. After arranging them, he found that some of them are left over. How many are left?

Concept & Strategy The problem asks us to find the largest perfect square less than or equal to $36581$. The difference between the total number of soldiers and this perfect square will be the number of soldiers left over. Step-by-Step Solution * Find the square root of $36581$ to determine the size of the largest possible square. * We know that $190^2 = 36100$, which is close to $36581$. * Let us check the next integer, $191$. $$191^2 = (190 + 1)^2 = 36100 + 380 + 1 = 36481$$ * Let us check $192$ just in case. $$192^2 = (190 + 2)^2 = 36100 + 760 + 4 = 36864$$ * Since $36864$ is greater than $36581$, the maximum number of soldiers in the square is $191 \times 191 = 36481$. * The number of soldiers left over = Total soldiers - Soldiers in the square $$\text{Left over} = 36581 - 36481 = 100$$ Exam Strategy & Shortcut To quickly approximate the square root, look at the first two digits ($36$) which suggests a base of $190$ (since $19^2 = 361$). Using the unit digit method, we test $191^2$ and $192^2$ to find the closest bound without going over. Common Pitfall Students might try to use the long division method for square roots and make an arithmetic error. In multiple-choice contexts, estimating and squaring numbers near the target is often faster and less error-prone. Final Answer **Therefore, the correct answer is 100.**