Let n be a positive integer. If a = (3^2n +4) ^ -1 , what is the remainder when a is divided by 9?

This question doesn't make a ton of sense.

a is clearly less than 1 and thus a divided by any integer m will be 0m + a,

i.e. a itself will be the remainder. Are you sure a isn't supposed to just be 3^{2n} + 4?

yes, I'm sure