Which is the largest number that divides `827, 1149 and 1310` to leave the same remainder in each case?

Options

A. 161
B. 116
C. 232
D. 322

Answer

To find the largest number that divides 827, 1149, and 1310 leaving the same remainder in each case, we can follow these steps:

### Step 1: Calculate the differences between the numbers
First, we need to find the differences between the numbers:

1. \( 1149 - 827 = 322 \)
2. \( 1310 - 1149 = 161 \)
3. \( 1310 - 827 = 483 \)

### Step 2: Find the HCF of the differences
Next, we need to find the highest common factor (HCF) of the differences we calculated: 322, 161, and 483.

### Step 3: Factorize the numbers
Now, we will factorize each of the differences:

1. **Factorizing 161**:
- \( 161 = 7 \times 23 \)

2. **Factorizing 322**:
- \( 322 = 2 \times 7 \times 23 \)

3. **Factorizing 483**:
- \( 483 = 3 \times 7 \times 23 \)

### Step 4: Identify common factors
From the factorizations, we can see that the common factors among 161, 322, and 483 are 7 and 23.

### Step 5: Calculate the HCF
To find the HCF, we take the product of the common factors:

\[ \text{HCF} = 7 \times 23 = 161 \]

### Conclusion
Thus, the largest number that divides 827, 1149, and 1310 leaving the same remainder is **161**.

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