Find the greatest number that divides 797, 1085 and 1232 and leaves 16,20,25 as remainders respectively.

Options

A. 69
B. 71
C. 65
D. 91

Answer

To find the greatest number that divides 797, 1085, and 1232, leaving remainders of 16, 20, and 25 respectively, we can follow these steps:

### Step 1: Adjust the Numbers
First, we need to adjust the given numbers by subtracting the respective remainders from each number.

- For 797, subtract 16:
\[ 797 - 16 = 781 \]

- For 1085, subtract 20:
\[ 1085 - 20 = 1065 \]

- For 1232, subtract 25:
\[ 1232 - 25 = 1207 \]

### Step 2: Find the HCF of the Adjusted Numbers
Now, we need to find the Highest Common Factor (HCF) of the adjusted numbers: 781, 1065, and 1207.

#### Finding HCF of 781 and 1065
1. **Prime Factorization of 781**:
- 781 = 11 × 71

2. **Prime Factorization of 1065**:
- 1065 = 15 × 71 = 3 × 5 × 71

3. **Common Factors**:
- The common factor is 71.

#### Finding HCF of 71 and 1207
1. **Prime Factorization of 1207**:
- 1207 = 17 × 71

2. **Common Factors**:
- The common factor remains 71.

### Step 3: Conclusion
The HCF of 781, 1065, and 1207 is 71. Therefore, the greatest number that divides 797, 1085, and 1232 and leaves the specified remainders is:

\[ \text{Greatest Number} = 71 \]

### Final Answer
The greatest number is **71**.

---