The greatest number which will divide 116, 221, 356 leaving the same remainder in each case is

Options

A. 15
B. 5
C. 10
D. 20

Answer

To find the greatest number that will divide 116, 221, and 356 leaving the same remainder in each case, we can follow these steps:

### Step 1: Define the numbers
Let:
- \( a = 116 \)
- \( b = 221 \)
- \( c = 356 \)

### Step 2: Calculate the differences
We need to find the differences between the numbers:
- \( b - a = 221 - 116 = 105 \)
- \( c - b = 356 - 221 = 135 \)
- \( c - a = 356 - 116 = 240 \)

### Step 3: Find the HCF of the differences
Now, we need to find the highest common factor (HCF) of the differences calculated:
- The differences are \( 105, 135, \) and \( 240 \).

### Step 4: Factor the differences
Let's factor each of the differences:
- \( 105 = 3 \times 5 \times 7 \)
- \( 135 = 3 \times 3 \times 5 \)
- \( 240 = 2^4 \times 3 \times 5 \)

### Step 5: Identify common factors
The common factors among the three numbers are:
- The common prime factors are \( 3 \) and \( 5 \).

### Step 6: Calculate the HCF
Now, we can calculate the HCF:
- HCF = \( 3 \times 5 = 15 \)

### Conclusion
Thus, the greatest number which will divide 116, 221, and 356 leaving the same remainder in each case is **15**.

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