When two numbers are separately divided by 33, the remainders are 21 and 28 respectively. If the sum of the two numbers is divided by 33, the remainder will be

Options

A. 10
B. 12
C. 14
D. 16

Answer

To solve the problem step by step, we can follow these instructions:

1. **Define the two numbers**: Let the two numbers be \( x \) and \( y \).

2. **Use the information given about remainders**:
- When \( x \) is divided by 33, the remainder is 21. This can be expressed mathematically as:
\[ x = 33k + 21 \quad \text{(for some integer } k\text{)} \]
- When \( y \) is divided by 33, the remainder is 28. This can be expressed as:
\[ y = 33m + 28 \quad \text{(for some integer } m\text{)} \]

3. **Find the sum of the two numbers**:
\[ x + y = (33k + 21) + (33m + 28) = 33k + 33m + 49 \]
Simplifying this gives:
\[ x + y = 33(k + m) + 49 \]

4. **Determine the remainder when \( x + y \) is divided by 33**:
- We can see that \( 33(k + m) \) is divisible by 33, so we only need to consider the remainder from the constant term:
\[ 49 \mod 33 \]
- To find \( 49 \mod 33 \), we can subtract 33 from 49:
\[ 49 - 33 = 16 \]

5. **Conclusion**:
- Therefore, the remainder when the sum of the two numbers \( x + y \) is divided by 33 is **16**.

### Summary of Steps:
1. Define the two numbers based on the given remainders.
2. Write equations for the two numbers using the divisors and remainders.
3. Sum the two equations.
4. Simplify to find the remainder when divided by 33.
5. Conclude with the final remainder.