Find the largest number which divides 1305, 4665 and 6905 leaving same remainder in each case.Also, find the common remainder.

Options

A. 1210,158
B. 1120,158
C. 1120,185
D. 1210,185

Answer

To find the largest number that divides 1305, 4665, and 6905 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 the numbers since the largest number that leaves the same remainder will divide these differences.

- Difference between 4665 and 1305:
\[ 4665 - 1305 = 3360 \]

- Difference between 6905 and 4665:
\[ 6905 - 4665 = 2240 \]

- Difference between 6905 and 1305:
\[ 6905 - 1305 = 5600 \]

### Step 2: Find the HCF of the differences
Now, we need to find the Highest Common Factor (HCF) of the differences calculated in Step 1: 3360, 2240, and 5600.

#### Finding HCF:
1. **Prime Factorization of each number:**
- **3360**:
\[ 3360 = 2^4 \times 3 \times 5 \times 7 \]
- **2240**:
\[ 2240 = 2^5 \times 5 \times 7 \]
- **5600**:
\[ 5600 = 2^4 \times 5^2 \times 7 \]

2. **Identify the common prime factors with the lowest powers:**
- For \(2\): minimum power is \(2^4\)
- For \(5\): minimum power is \(5^1\)
- For \(7\): minimum power is \(7^1\)

3. **Calculate the HCF:**
\[ HCF = 2^4 \times 5^1 \times 7^1 = 16 \times 5 \times 7 = 560 \]

### Step 3: Find the common remainder
To find the common remainder, we can use any of the original numbers and subtract the HCF from it.

Using 1305:
\[ \text{Common Remainder} = 1305 - 560 \times k \]
where \(k\) is an integer such that the result is non-negative.

Calculating for \(k=2\):
\[ 1305 - 560 \times 2 = 1305 - 1120 = 185 \]

### Final Answer:
- The largest number that divides 1305, 4665, and 6905 leaving the same remainder is **560**.
- The common remainder is **185**.