Explanation:

- To find the last digit of \( S = 9^{27} + 27^9 \), we need to determine the last digit of each term separately.

- For \( 9^{27} \):

- The last digits of powers of 9 repeat every two cycles: 9, 1.

- Since 27 is odd, the last digit of \( 9^{27} \) is 9.

- For \( 27^9 \):

- The last digits of powers of 7 follow a repetitive pattern: 7, 9, 3, 1.

- Since \( 9 \mod 4 = 1 \), the last digit of \( 27^9 \) is the same as that of \( 7^1 \), which is 7.

- Adding both results: \( 9 + 7 = 16 \).

- The last digit of this sum is 6.

- Answer: Option 2 – 6