What is the last digit of the sum S = 9²⁷ + 27⁹ ?

Concept:

The last digit of a number that is being raised to a power often follows a cycle.

For example, the last digit of 9ⁿ repeats every 2 numbers (9, 1, 9, 1, ...).

Similarly, 7ⁿ repeats every 4 numbers (7, 9, 3, 1, 7, 9, 3, 1, ...).

Calculation:

For 9ⁿ, the last digit is 9 if n is odd and 1 if n is even.

In this case, 9²⁷ has a last digit of 9 because 27 is odd.

For 27⁹, we only need to consider the last digit of 27, which is 7.

To find the last digit of 27⁹, we need to find the remainder when 9 is divided

by 4 (because the cycle length is 4). That remainder is 1.

 So the last digit of 27⁹ is the first digit in the cycle, which is 7.

Now

9 (from 9²⁷) + 7 (from 27⁹) = 16.

∴ The last digit of the sum S = 9²⁷ + 27⁹ is 6.