1960 AHSME 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

\begin{array}{l} \textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad \\ \textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad \\ \textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad \\ \textbf{(E)}\ \text{II can be solved in consecutive odd integers} \end{array}

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