1992 AHSME Problem 4

If a,b and c are positive integers and a and b are odd, then 3^a+(b-1)^2c is

\begin{array}{l} \text{(A) odd for all choices of c} \quad \text{(B) even for all choices of c} \quad\\ \text{(C) odd if c is even; even if c is odd} \quad\\ \text{(D) odd if c is odd; even if c is even} \quad\\ \text{(E) odd if c is not a multiple of 3; even if c is a multiple of 3} \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.