AHSME 1960 Problem 19

Consider equation $I: x+y+z=46$ where $x, y$, and $z$ are positive integers, and equation $II: x+y+z+w=46$, where $x, y, z$, and $w$ are positive integers. Then $\textbf{(A)}\ \text{I can be solved in consecutive integers} \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \\ \textbf{(C)}\ \text{II can be solved in consecutive integers} \\ \textbf{(D)}\ \text{II can be solved in consecutive even integers} \\ \textbf{(E)}\ \text{II can be solved in consecutive odd integers}$