The greatest number that divides 411, 684, 821 and leaves 3, 4 and 5 as remainders, respectively, is:

Options

A. 245
B. 146
C. 136
D. 204

Answer

To find the greatest number that divides 411, 684, and 821 while leaving remainders of 3, 4, and 5 respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Adjust the Numbers by Subtracting Remainders**:
- Subtract the remainders from each of the numbers:
- For 411, subtract 3:
\( 411 - 3 = 408 \)
- For 684, subtract 4:
\( 684 - 4 = 680 \)
- For 821, subtract 5:
\( 821 - 5 = 816 \)

2. **Find the HCF (Highest Common Factor)**:
- Now, we need to find the HCF of the adjusted numbers: 408, 680, and 816.

3. **Prime Factorization**:
- Factor each number into its prime factors:
- **408**:
\( 408 = 2^3 \times 3 \times 17 \)
- **680**:
\( 680 = 2^3 \times 5 \times 17 \)
- **816**:
\( 816 = 2^4 \times 3 \times 17 \)

4. **Identify Common Factors**:
- The common prime factors among the three numbers are:
- \( 2^3 \) (the minimum power of 2 among the three)
- \( 17 \) (common in all three)

5. **Calculate the HCF**:
- Multiply the common factors:
- \( HCF = 2^3 \times 17 = 8 \times 17 = 136 \)

6. **Conclusion**:
- The greatest number that divides 411, 684, and 821 and leaves the respective remainders is **136**.