Find the largest number which divides 55,127 and 175 so as to leave the same remainder in each case .

Answer

To find the largest number that divides 55, 127, and 175 leaving the same remainder in each case, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find a number \( x \) such that when we divide 55, 127, and 175 by \( x \), the remainders are the same. This means that the differences between the numbers should be divisible by \( x \).

2. **Calculate the Differences**:
We calculate the differences between the numbers:
- \( 127 - 55 = 72 \)
- \( 175 - 127 = 48 \)
- \( 175 - 55 = 120 \)

3. **List the Differences**:
The differences we found are:
- 72
- 48
- 120

4. **Finding the HCF**:
Now, we need to find the highest common factor (HCF) of these differences: 72, 48, and 120.

5. **Prime Factorization**:
- **For 72**:
\[ 72 = 2^3 \times 3^2 \]
- **For 48**:
\[ 48 = 2^4 \times 3^1 \]
- **For 120**:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

6. **Finding the Common Factors**:
We take the lowest power of each prime factor that appears in all three factorizations:
- For \( 2 \): The minimum power is \( 2^3 \).
- For \( 3 \): The minimum power is \( 3^1 \).
- \( 5 \) is not common in all three.

7. **Calculating the HCF**:
Therefore, the HCF is:
\[ HCF = 2^3 \times 3^1 = 8 \times 3 = 24 \]

8. **Conclusion**:
The largest number that divides 55, 127, and 175 leaving the same remainder is **24**.