I would use induction, observing that:
even simpler:
9 = 4 (mod 5), whence 9^{3} = 4^{3} = 4 (mod 5) (since 4^{2} = 16 = 1 (mod 5)).
thus 9^{n+3} + 4^{n} = (4^{n})4 + 4^{n} = 5(4^{n}) = 0 (mod 5).
why do this? because why should i have to calculate the cube of 9, when i can calculate the cube of 4 instead (729 is a number i don't use everyday)?