What is the largest number which when divides 1475, 3155 and 5255 leave the same remainder in each case ?

Options

A. 320
B. 420
C. 350
D. 410

Answer

To find the largest number that divides 1475, 3155, and 5255 leaving the same remainder in each case, we can follow these steps:

### Step 1: Find the differences between the numbers
We need to calculate the differences between the given numbers:
- Difference 1: \( 3155 - 1475 = 1680 \)
- Difference 2: \( 5255 - 3155 = 2100 \)
- Difference 3: \( 5255 - 1475 = 3780 \)

### Step 2: List the differences
Now we have the following differences:
- \( 1680 \)
- \( 2100 \)
- \( 3780 \)

### Step 3: Factor each difference
Next, we will factor each of these differences:
- **For 1680**:
- \( 1680 = 2^4 \times 3 \times 5 \times 7 \)

- **For 2100**:
- \( 2100 = 2^2 \times 3 \times 5^2 \times 7 \)

- **For 3780**:
- \( 3780 = 2^2 \times 3^3 \times 5 \times 7 \)

### Step 4: Find the HCF of the differences
Now we need to find the Highest Common Factor (HCF) of the three factored differences:
- From the factorizations:
- For \( 2 \): The minimum power is \( 2^2 \)
- For \( 3 \): The minimum power is \( 3^1 \)
- For \( 5 \): The minimum power is \( 5^1 \)
- For \( 7 \): The minimum power is \( 7^1 \)

Thus, the HCF is:
\[ HCF = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

### Step 5: Calculate the HCF
Now we calculate the HCF:
\[ HCF = 4 \times 3 \times 5 \times 7 \]
Calculating step-by-step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 5 = 60 \)
- \( 60 \times 7 = 420 \)

### Conclusion
The largest number which divides 1475, 3155, and 5255 leaving the same remainder is **420**.

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