There are 576 boys and 448 girls in a school that are to be divided into equal sections of either boys or girls alone . Find the total number of sections thus formed.

Answer

To solve the problem, we need to find the total number of sections formed by dividing 576 boys and 448 girls into equal sections of either boys or girls alone. We will do this by finding the Highest Common Factor (HCF) of the two numbers.

### Step-by-Step Solution:

1. **Identify the numbers of boys and girls:**
- Number of boys = 576
- Number of girls = 448

2. **Find the prime factorization of 576:**
- Divide 576 by 2:
- 576 ÷ 2 = 288
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Thus, the prime factorization of 576 is:
\[ 576 = 2^6 \times 3^2 \]

3. **Find the prime factorization of 448:**
- Divide 448 by 2:
- 448 ÷ 2 = 224
- 224 ÷ 2 = 112
- 112 ÷ 2 = 56
- 56 ÷ 2 = 28
- 28 ÷ 2 = 14
- 14 ÷ 2 = 7
- 7 ÷ 7 = 1
- Thus, the prime factorization of 448 is:
\[ 448 = 2^6 \times 7^1 \]

4. **Find the HCF of 576 and 448:**
- The common prime factors are \(2\).
- The minimum power of \(2\) in both factorizations is \(2^6\).
- Therefore, the HCF is:
\[ HCF = 2^6 = 64 \]

5. **Calculate the number of sections for boys:**
- Number of sections for boys = Total boys ÷ HCF
\[ \text{Number of boy sections} = \frac{576}{64} = 9 \]

6. **Calculate the number of sections for girls:**
- Number of sections for girls = Total girls ÷ HCF
\[ \text{Number of girl sections} = \frac{448}{64} = 7 \]

7. **Calculate the total number of sections formed:**
- Total sections = Number of boy sections + Number of girl sections
\[ \text{Total sections} = 9 + 7 = 16 \]

### Final Answer:
The total number of sections formed is **16**.