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 of dividing 576 boys and 448 girls into equal sections, we will follow these steps:

### Step 1: Find the HCF (Highest Common Factor) of 576 and 448
To divide the boys and girls into equal sections, we need to find the HCF of the two numbers.

#### Finding the prime factorization:
- **576**:
- Divide by 2: 576 ÷ 2 = 288
- Divide by 2: 288 ÷ 2 = 144
- Divide by 2: 144 ÷ 2 = 72
- Divide by 2: 72 ÷ 2 = 36
- Divide by 2: 36 ÷ 2 = 18
- Divide by 2: 18 ÷ 2 = 9
- Divide by 3: 9 ÷ 3 = 3
- Divide by 3: 3 ÷ 3 = 1
- Therefore, the prime factorization of 576 is \( 2^6 \times 3^2 \).

- **448**:
- Divide by 2: 448 ÷ 2 = 224
- Divide by 2: 224 ÷ 2 = 112
- Divide by 2: 112 ÷ 2 = 56
- Divide by 2: 56 ÷ 2 = 28
- Divide by 2: 28 ÷ 2 = 14
- Divide by 2: 14 ÷ 2 = 7
- Divide by 7: 7 ÷ 7 = 1
- Therefore, the prime factorization of 448 is \( 2^6 \times 7^1 \).

#### Finding the HCF:
- The common prime factors are \( 2^6 \).
- Thus, the HCF of 576 and 448 is \( 2^6 = 64 \).

### Step 2: Calculate the number of sections for boys and girls
Now, we will divide the total number of boys and girls by the HCF to find the number of sections.

- **Number of sections for boys**:
\[ \text{Number of sections for boys} = \frac{576}{64} = 9 \]

- **Number of sections for girls**:
\[ \text{Number of sections for girls} = \frac{448}{64} = 7 \]

### Step 3: Calculate the total number of sections
Now, we will add the number of sections for boys and girls to find the total number of sections.

\[ \text{Total number of sections} = 9 + 7 = 16 \]

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

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