A P.T. teacher asks 20736 students to stand in such a way that there are as many rows as there are students in each row. Find the number of rows.

Options

A. 124
B. 126
C. 134
D. 144

Answer

To solve the problem, we need to find the number of rows (N) such that the number of students in each row is also N. This means that the total number of students (20736) can be expressed as the product of the number of rows and the number of students in each row, which is:

\[ N \times N = N^2 = 20736 \]

To find N, we need to take the square root of 20736:

### Step 1: Calculate the square root of 20736
\[ N = \sqrt{20736} \]

### Step 2: Perform the calculation
To find the square root, we can either use a calculator or factor the number. Let's factor it:

1. Start by dividing 20736 by 2 (the smallest prime number):
- 20736 ÷ 2 = 10368
- 10368 ÷ 2 = 5184
- 5184 ÷ 2 = 2592
- 2592 ÷ 2 = 1296
- 1296 ÷ 2 = 648
- 648 ÷ 2 = 324
- 324 ÷ 2 = 162
- 162 ÷ 2 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 20736 is:
\[ 20736 = 2^8 \times 3^4 \]

### Step 3: Use the prime factorization to find the square root
To find the square root, we take half the powers of the prime factors:
\[ \sqrt{20736} = \sqrt{2^8 \times 3^4} = 2^{8/2} \times 3^{4/2} = 2^4 \times 3^2 \]

Calculating this gives:
\[ 2^4 = 16 \]
\[ 3^2 = 9 \]
\[ 16 \times 9 = 144 \]

### Conclusion
Thus, the number of rows (N) is:
\[ N = 144 \]

### Final Answer
The number of rows is **144**.

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