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 need to find the remainder when the sum of two numbers is divided by 33, given their individual remainders when divided by 33.

### Step-by-Step Solution:

1. **Identify the two numbers and their remainders**:
- Let the two numbers be \( n_1 \) and \( n_2 \).
- When \( n_1 \) is divided by 33, the remainder is 21.
- When \( n_2 \) is divided by 33, the remainder is 28.

2. **Express the two numbers in terms of their remainders**:
- We can express \( n_1 \) and \( n_2 \) as:
\[ n_1 = 33p + 21 \]
\[ n_2 = 33q + 28 \]
where \( p \) and \( q \) are the quotients when \( n_1 \) and \( n_2 \) are divided by 33, respectively.

3. **Find the sum of the two numbers**:
- Now, we can find the sum \( n_1 + n_2 \):
\[ n_1 + n_2 = (33p + 21) + (33q + 28) \]
\[ n_1 + n_2 = 33p + 33q + 21 + 28 \]
\[ n_1 + n_2 = 33(p + q) + 49 \]

4. **Determine the remainder when the sum is divided by 33**:
- Now, we need to find the remainder when \( n_1 + n_2 \) is divided by 33:
\[ n_1 + n_2 = 33(p + q) + 49 \]
- The term \( 33(p + q) \) is divisible by 33, so we only need to consider the remainder of 49 when divided by 33.

5. **Calculate \( 49 \mod 33 \)**:
- To find the remainder, we perform the division:
\[ 49 \div 33 = 1 \quad \text{(quotient)} \]
\[ 49 - 33 \times 1 = 49 - 33 = 16 \quad \text{(remainder)} \]

6. **Conclusion**:
- Therefore, the remainder when the sum of the two numbers is divided by 33 is **16**.

### Final Answer:
The remainder is **16**.