765 chairs are to be arranged in the columns in such a way that the number of chairs in each column should be equal to the total number of columns. How many chairs will be excluded to make this arrangement possible?

Options

A. 12
B. 24
C. 36
D. 48

Answer

To solve the problem, we need to arrange 765 chairs in such a way that the number of chairs in each column is equal to the total number of columns. Let's denote the number of columns as \( n \). Therefore, the number of chairs in each column will also be \( n \).

This means the total number of chairs can be expressed as:
\[ n \times n = n^2 \]
We need to find the largest perfect square less than or equal to 765, because we want to maximize the number of chairs used in the arrangement.

### Step 1: Find the largest perfect square less than or equal to 765.
To find this, we can take the square root of 765:
\[ \sqrt{765} \approx 27.7 \]
This means the largest integer \( n \) that satisfies \( n^2 \leq 765 \) is 27.

### Step 2: Calculate \( n^2 \) for \( n = 27 \).
Now we calculate:
\[ 27^2 = 729 \]
This is the largest perfect square that is less than 765.

### Step 3: Determine how many chairs will be excluded.
Now we need to find out how many chairs will be excluded:
\[ \text{Chairs excluded} = 765 - 729 \]
Calculating this gives:
\[ \text{Chairs excluded} = 36 \]

### Conclusion:
Thus, the number of chairs that will be excluded to make this arrangement possible is **36**.

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