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
$\textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad \\ \textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad \\ \textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad \\ \textbf{(E)}\ \text{II can be solved in consecutive odd integers}$
Consider each option, one at a time.
For option A, let $x=y-1$ and $z=y+1$.  That means $3y=46$, so $y=\frac{46}{3}$.  That is not an integer, so option A is eliminated.
For option B, let $x=y-2$ and $z=y+2$.  That also means $3y=46$, so $y=\frac{46}{3}$.  That is also not an integer, so option B is eliminated.
For option C, let $y=x+1$, $z=x+2$, and $w=x+3$.  That means $4x+6=46$, so $x=10$.  Option C works.
For options D and E, let $y=x+2$, $z=x+4$, and $w=x+6$.  That means $4x+12=46$, so $x=\frac{17}{2}$.  Since the result is not an integer, options D and E are eliminated.
Thus, the answer is $\boxed{\textbf{(C)}}$.