In a school, 391 boys and 323 girls have been divided into the largest possible equal classes , so that each class of boys numbers the same as each class of girls. What is the number of classes ?

Options

A. 23
B. 19
C. 44
D. 17

Answer

To solve the problem of dividing 391 boys and 323 girls into the largest possible equal classes, we need to find the Highest Common Factor (HCF) of the two numbers. Here’s a step-by-step solution:

### Step 1: Factorize the Numbers
First, we need to factorize both 391 and 323 into their prime factors.

- **For 391**:
- We can check for divisibility by prime numbers.
- 391 is not divisible by 2 (it's odd).
- Dividing by 3: 3 + 9 + 1 = 13 (not divisible).
- Dividing by 5: last digit is not 0 or 5.
- Dividing by 7: 391 ÷ 7 = 55.857 (not divisible).
- Dividing by 17: 391 ÷ 17 = 23 (391 = 17 × 23).

So, the prime factorization of 391 is:
\[ 391 = 17 \times 23 \]

- **For 323**:
- Checking for divisibility by prime numbers.
- 323 is not divisible by 2 (it's odd).
- Dividing by 3: 3 + 2 + 3 = 8 (not divisible).
- Dividing by 5: last digit is not 0 or 5.
- Dividing by 7: 323 ÷ 7 = 46.142 (not divisible).
- Dividing by 17: 323 ÷ 17 = 19 (323 = 17 × 19).

So, the prime factorization of 323 is:
\[ 323 = 17 \times 19 \]

### Step 2: Identify the Common Factors
Now, we look for the common factors in the prime factorizations of both numbers:
- Factors of 391: 17, 23
- Factors of 323: 17, 19

The common factor is:
\[ \text{Common Factor} = 17 \]

### Step 3: Determine the Number of Classes
The Highest Common Factor (HCF) gives us the largest number of equal classes that can be formed. Therefore, the number of classes is equal to the HCF:
\[ \text{Number of Classes} = \text{HCF}(391, 323) = 17 \]

### Conclusion
Thus, the number of classes is **17**.

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