$\displaystyle 2^{a+b}\equiv 2^b+3 \mod9$

I did a case analysis on b mod 6 and found the congruences a has to satisfy, but I don't think my answer is right... Any good resources on modular arithmetic are appreciated also!

EDIT: Sorry, clarification. I am looking for necessary and sufficient conditions that a and b satisfy the congruence. Like I said, I broke into cases on a and b mod 6 and got as my answer

$\displaystyle a\equiv 2, b\equiv 0,2,4 \mod 6$

$\displaystyle a\equiv 4, b\equiv 1,3,5 \mod 6$

but the process is ugly and I don't think I'm right - -;