The greatest number which can divide 1354 , 1866 and 2762 leaving the same remainder 10 in each case is -

Options

A. 64
B. 124
C. 156
D. 260

Answer

To find the greatest number that can divide 1354, 1866, and 2762 leaving the same remainder of 10 in each case, we can follow these steps:

### Step 1: Subtract the remainder from each number
We first need to subtract the remainder (10) from each of the given numbers:

- For 1354:
\[ 1354 - 10 = 1344 \]
- For 1866:
\[ 1866 - 10 = 1856 \]
- For 2762:
\[ 2762 - 10 = 2752 \]

### Step 2: Find the HCF of the adjusted numbers
Next, we need to find the Highest Common Factor (HCF) of the numbers obtained in Step 1: 1344, 1856, and 2752.

### Step 3: Prime factorization of each number
Now, we will perform the prime factorization of each number:

1. **Prime factorization of 1344**:
\[ 1344 = 2^6 \times 3^1 \times 7^1 \]

2. **Prime factorization of 1856**:
\[ 1856 = 2^7 \times 29^1 \]

3. **Prime factorization of 2752**:
\[ 2752 = 2^6 \times 43^1 \]

### Step 4: Identify the common factors
Now, we will identify the common factors from the prime factorizations:

- The common prime factor is \(2\).
- The minimum power of \(2\) in the factorizations is \(2^6\).

### Step 5: Calculate the HCF
Thus, the HCF is:
\[ HCF = 2^6 = 64 \]

### Conclusion
The greatest number which can divide 1354, 1866, and 2762 leaving the same remainder of 10 in each case is **64**.

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