This was a good one.

First, we need to figure a few powers of 9 to determine a pattern:
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = 6561

We can tell that 9 raised to an odd power will produce a number ending in 9, while 9 raised to an even number will produce a number ending in 1.

12n + 3 where n > 0 will always produce an odd number, so 9^(12n+3) will always end in 9.

If we divide 729, for example, by 10 we get remainder 9. Hence, we will have a remainder of 9.

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