The largest number which divides 77, 147, and 252 to leave the same remainder in each case is-

Options

A. 9
B. 15
C. 25
D. 35

Answer

To find the largest number that divides 77, 147, and 252 leaving the same remainder in each case, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Numbers**: We have the numbers 77, 147, and 252.

2. **Calculate the Differences**:
- Find the differences between the numbers:
- Difference between 147 and 77:
\[ 147 - 77 = 70 \]
- Difference between 252 and 147:
\[ 252 - 147 = 105 \]
- Difference between 252 and 77:
\[ 252 - 77 = 175 \]
- So, the differences are 70, 105, and 175.

3. **Find the HCF of the Differences**:
- We need to find the highest common factor (HCF) of the differences 70, 105, and 175.
- Let's factor each number:
- \(70 = 2 \times 5 \times 7\)
- \(105 = 3 \times 5 \times 7\)
- \(175 = 5 \times 5 \times 7\)
- Now, identify the common factors:
- The common factors are \(5\) and \(7\).

4. **Calculate the HCF**:
- Multiply the common factors:
\[ HCF = 5 \times 7 = 35 \]

5. **Conclusion**:
- The largest number which divides 77, 147, and 252 leaving the same remainder is **35**.