23. If the zeros of the function $f(x)=x^{2}-a x+2 a$ are integers, then the sum of all possible values of the coefficient $a$ is ( ).
(A) 7
(B) 8
(C) 16
(D) 17
(E) 18
23. C.

Since the zeros of the function $f(x)=x^{2}-a x+2 a$ are integers, according to the problem and Vieta's formulas, we know that $a$ is also an integer, and the discriminant of the quadratic function $f(x)=x^{2}-a x+2 a$ is a perfect square.
$$
\begin{array}{l}
\text { Let } \Delta=a^{2}-8 a=k^{2}(k \in \mathbf{N}) \text {. Then } \\
a^{2}-8 a+16=k^{2}+16 \\
\Rightarrow(a-4)^{2}-k^{2}=16 \\
\Rightarrow(a-4+k)(a-4-k)=16 .
\end{array}
$$

Since $a-4+k$ and $a-4-k$ have the same parity, we have
$$
\begin{array}{l}
(a-4+k, a-4-k) \\
=(8,2),(4,4),(-4,-4),(-2,-8) .
\end{array}
$$

Solving these, we get $a_{1}=9, a_{2}=8, a_{3}=0, a_{4}=-1$.
Thus, the sum of all possible values of the coefficient $a$ is 16.