`9^11 + 11^9` is divisible by

Options

A. a)7
B. b) 8
C. c) 9
D. d) 10

Answer

To determine the divisibility of \(9^{11} + 11^{9}\), we can analyze the last digits of each term separately and then combine them.

### Step-by-Step Solution:

1. **Find the last digit of \(9^{11}\)**:
- The last digit of powers of 9 follows a pattern:
- \(9^1 = 9\) (last digit is 9)
- \(9^2 = 81\) (last digit is 1)
- \(9^3 = 729\) (last digit is 9)
- \(9^4 = 6561\) (last digit is 1)
- The pattern alternates between 9 and 1. Since \(11\) is odd, the last digit of \(9^{11}\) is **9**.

2. **Find the last digit of \(11^{9}\)**:
- The last digit of powers of 11 is always 1:
- \(11^1 = 11\) (last digit is 1)
- \(11^2 = 121\) (last digit is 1)
- \(11^3 = 1331\) (last digit is 1)
- Therefore, the last digit of \(11^{9}\) is **1**.

3. **Add the last digits**:
- Now, we add the last digits obtained:
\[ 9 + 1 = 10 \]
- The last digit of the sum \(9^{11} + 11^{9}\) is **0**.

4. **Conclusion about divisibility**:
- Since the last digit of \(9^{11} + 11^{9}\) is 0, it is divisible by **10**.

### Final Answer:
Thus, \(9^{11} + 11^{9}\) is divisible by **10**.

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