The expression `2222^(7777) +7777^(2222)` is divisible by :

Options

A. 99
B. 101
C. 13
D. any two of these

Answer

To determine the divisibility of the expression \( 2222^{7777} + 7777^{2222} \), we will analyze the numbers involved and check their divisibility by the given options.

### Step-by-Step Solution:

1. **Identify the Numbers**:
- We have two numbers: \( 2222 \) and \( 7777 \).

2. **Check Divisibility by 101**:
- First, we note that \( 2222 \) can be expressed as \( 22 \times 101 \) (since \( 2222 = 2 \times 1000 + 2 \times 200 + 2 \times 20 + 2 \times 2 \)).
- Similarly, \( 7777 \) can be expressed as \( 77 \times 101 \).
- Since both \( 2222 \) and \( 7777 \) are multiples of \( 101 \), any power of these numbers will also be divisible by \( 101 \).
- Therefore, \( 2222^{7777} \) is divisible by \( 101 \) and \( 7777^{2222} \) is also divisible by \( 101 \).
- Thus, their sum \( 2222^{7777} + 7777^{2222} \) is divisible by \( 101 \).

3. **Check Divisibility by 99**:
- A number is divisible by \( 99 \) if it is divisible by both \( 9 \) and \( 11 \).
- **Divisibility by 11**:
- For \( 2222 \), the sum of the digits is \( 2 + 2 + 2 + 2 = 8 \) (which is not divisible by \( 11 \)).
- For \( 7777 \), the sum of the digits is \( 7 + 7 + 7 + 7 = 28 \) (which is also not divisible by \( 11 \)).
- Therefore, \( 2222 \) and \( 7777 \) are not divisible by \( 11 \).
- Since neither number is divisible by \( 11 \), their sum cannot be divisible by \( 99 \).

4. **Check Divisibility by 13**:
- We can check the divisibility of \( 2222 \) and \( 7777 \) by \( 13 \).
- \( 2222 \div 13 \approx 171.692 \) (not divisible).
- \( 7777 \div 13 \approx 599.0 \) (divisible).
- Since \( 2222 \) is not divisible by \( 13 \), their sum cannot be divisible by \( 13 \).

5. **Conclusion**:
- The expression \( 2222^{7777} + 7777^{2222} \) is divisible by \( 101 \) but not by \( 99 \) or \( 13 \).
- Therefore, the correct answer is that the expression is divisible by \( 101 \).

### Final Answer:
The expression \( 2222^{7777} + 7777^{2222} \) is divisible by \( 101 \).