A farmer has 945 cows and 2475 sheep. He farms them into flock keeping cows and sheep separate and having the same number of animals in each flock. If these flocks are as large as possible, then the maximum number of animals in each flock and total number of flock required for the purpose are respectively.

Options

A. 15 and 228
B. 9 and 380
C. 45 and 76
D. 46 and 75

Answer

To solve the problem, we need to find the maximum number of animals in each flock, which is determined by finding the Highest Common Factor (HCF) of the number of cows and sheep. Then, we will calculate the total number of flocks required for both cows and sheep.

### Step-by-Step Solution:

1. **Identify the numbers**:
- Number of cows = 945
- Number of sheep = 2475

2. **Find the HCF of 945 and 2475**:
- To find the HCF, we can use the prime factorization method or the division method. Here, we will use the prime factorization method.

- **Prime Factorization of 945**:
- 945 = 3 × 315
- 315 = 3 × 105
- 105 = 3 × 35
- 35 = 5 × 7
- Therefore, the prime factorization of 945 is:
\[ 945 = 3^3 × 5^1 × 7^1 \]

- **Prime Factorization of 2475**:
- 2475 = 5 × 495
- 495 = 5 × 99
- 99 = 3 × 33
- 33 = 3 × 11
- Therefore, the prime factorization of 2475 is:
\[ 2475 = 3^2 × 5^2 × 11^1 \]

- **Finding the HCF**:
- The HCF is found by taking the lowest power of all prime factors common to both numbers.
- Common prime factors:
- For 3: minimum power is \(3^2\)
- For 5: minimum power is \(5^1\)
- Thus, the HCF is:
\[ HCF = 3^2 × 5^1 = 9 × 5 = 45 \]

3. **Determine the maximum number of animals in each flock**:
- The maximum number of animals in each flock is the HCF we calculated, which is 45.

4. **Calculate the total number of flocks required**:
- For cows:
\[ \text{Number of flocks for cows} = \frac{945}{45} = 21 \]
- For sheep:
\[ \text{Number of flocks for sheep} = \frac{2475}{45} = 55 \]

5. **Total number of flocks**:
- Total flocks = Number of flocks for cows + Number of flocks for sheep
\[ \text{Total flocks} = 21 + 55 = 76 \]

### Final Answer:
- The maximum number of animals in each flock is **45**.
- The total number of flocks required is **76**.