Find the greatest number that will divide 55, 127 and 175, so as to leave the same remainder in each case

Options

A. 1
B. 16
C. 24
D. 16

Answer

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

### Step 1: Calculate the differences between the numbers
We need to find the differences between each pair of numbers:

1. \( 127 - 55 = 72 \)
2. \( 175 - 127 = 48 \)
3. \( 175 - 55 = 120 \)

### Step 2: Find the HCF of the differences
Now, we need to find the Highest Common Factor (HCF) of the differences we calculated: 72, 48, and 120.

### Step 3: Prime factorization of each difference
1. **For 72**:
- \( 72 = 2^3 \times 3^2 \)

2. **For 48**:
- \( 48 = 2^4 \times 3^1 \)

3. **For 120**:
- \( 120 = 2^3 \times 3^1 \times 5^1 \)

### Step 4: Identify the common factors
Now, we find the common prime factors from the factorizations:
- The minimum power of \(2\) present in all three numbers is \(2^3\).
- The minimum power of \(3\) present in all three numbers is \(3^1\).

### Step 5: Calculate the HCF
Now we can calculate the HCF:
\[ \text{HCF} = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Conclusion
The greatest number that will divide 55, 127, and 175, leaving the same remainder, is **24**.

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