The largest number which will divide the numbers 104, 221 and 377 leaving the same remainder in each case is :

Options

A. 56
B. 13
C. 39
D. it does not exist

Answer

To find the largest number that will divide the numbers 104, 221, and 377 leaving the same remainder, we can follow these steps:

### Step 1: Calculate the differences between the numbers
We need to find the differences between the pairs of numbers:
- Difference between 221 and 104:
\[ 221 - 104 = 117 \]
- Difference between 377 and 221:
\[ 377 - 221 = 156 \]
- Difference between 377 and 104:
\[ 377 - 104 = 273 \]

### Step 2: Find the Greatest Common Divisor (GCD) of the differences
Now we need to find the GCD of the three differences calculated:
- The differences are 117, 156, and 273.

### Step 3: Prime factorization of each difference
1. **For 117**:
- \(117 = 3 \times 39\)
- \(39 = 3 \times 13\)
- So, \(117 = 3^2 \times 13\)

2. **For 156**:
- \(156 = 2 \times 78\)
- \(78 = 2 \times 39\)
- \(39 = 3 \times 13\)
- So, \(156 = 2^2 \times 3 \times 13\)

3. **For 273**:
- \(273 = 3 \times 91\)
- \(91 = 7 \times 13\)
- So, \(273 = 3 \times 7 \times 13\)

### Step 4: Identify common factors
Now we need to identify the common prime factors from the factorizations:
- From \(117\): \(3^2, 13\)
- From \(156\): \(2^2, 3, 13\)
- From \(273\): \(3, 7, 13\)

The common factors among all three numbers are \(3\) and \(13\).

### Step 5: Calculate the GCD
The GCD is found by taking the lowest power of all common prime factors:
- For \(3\), the lowest power is \(3^1\).
- For \(13\), the lowest power is \(13^1\).

Thus, the GCD is:
\[ GCD = 3^1 \times 13^1 = 3 \times 13 = 39 \]

### Conclusion
The largest number which will divide the numbers 104, 221, and 377 leaving the same remainder is **39**.

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