What will be the greatest number that divides 1356, 1868 and 2764 leaving 12 as remainder in each case?

Options

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

Answer

To find the greatest number that divides 1356, 1868, and 2764 leaving a remainder of 12 in each case, we can follow these steps:

### Step 1: Subtract the remainder from each number
To find the numbers that are fully divisible by the greatest number, we need to subtract the remainder (12) from each of the given numbers.

- For 1356:
\[ 1356 - 12 = 1344 \]

- For 1868:
\[ 1868 - 12 = 1856 \]

- For 2764:
\[ 2764 - 12 = 2752 \]

### Step 2: Find the HCF of the resulting numbers
Now, we need to find the highest common factor (HCF) of the three resulting numbers: 1344, 1856, and 2752.

### Step 3: Prime factorization of each number
1. **Prime factorization of 1344**:
- 1344 = 2 × 672
- 672 = 2 × 336
- 336 = 2 × 168
- 168 = 2 × 84
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- So, \( 1344 = 2^6 × 3^1 × 7^1 \)

2. **Prime factorization of 1856**:
- 1856 = 2 × 928
- 928 = 2 × 464
- 464 = 2 × 232
- 232 = 2 × 116
- 116 = 2 × 58
- 58 = 2 × 29
- So, \( 1856 = 2^5 × 29^1 \)

3. **Prime factorization of 2752**:
- 2752 = 2 × 1376
- 1376 = 2 × 688
- 688 = 2 × 344
- 344 = 2 × 172
- 172 = 2 × 86
- 86 = 2 × 43
- So, \( 2752 = 2^5 × 43^1 \)

### Step 4: Identify common factors
Now, we will identify the common prime factors from the factorizations:
- The only common prime factor is \( 2 \).
- The minimum power of 2 in all three factorizations is \( 2^5 \).

### Step 5: Calculate the HCF
Thus, the HCF of 1344, 1856, and 2752 is:
\[ HCF = 2^5 = 32 \]

### Step 6: Conclusion
The greatest number that divides 1356, 1868, and 2764 leaving a remainder of 12 in each case is:
\[ \text{Answer} = 32 \]

---