5. On a Sunday, ninth-grade student Xiao Hua went to the market to buy vegetables for his mother at a little past 6 o'clock in the morning. Before leaving, he glanced at his watch and noticed that the angle between the hour and minute hands was $\alpha^{\circ}$ (where $\alpha$ is an integer). When he returned, he also checked his watch and found that it was not yet 7 o'clock, and the angle between the hour and minute hands was also $\alpha^{\circ}$. If Xiao Hua spent $10 k$ (where $k$ is an integer) minutes buying vegetables, then the number of possible values for $k$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
5. B.

Let the angle between the hour hand and the minute hand be $\alpha$ when it is $n$ o'clock and $m$ minutes. Then, when the hour hand is ahead,
$$
\alpha=30 n-6 m+0.5 m=30 n-5.5 m \text {; }
$$

When the minute hand is ahead, $\alpha=5.5 m-30 n$.
Suppose the time of departure is 6 o'clock and $x$ minutes, and the time of return is 6 o'clock and $y$ minutes.
Then the time spent buying vegetables is $y-x$ minutes, and
$$
\begin{array}{l}
\alpha=30 \times 6-5.5 x, \\
\alpha=5.5 y-30 \times 6 . \\
\text { (1) }+ \text { (2) gives } \\
2 \alpha=5.5(y-x) \Rightarrow y-x=\frac{2 \alpha}{5.5}=\frac{4 \alpha}{11} .
\end{array}
$$

Since $y-x=10 k$, we have
$$
\frac{4 \alpha}{11}=10 k \Rightarrow 2 \alpha=55 k \text {. }
$$

Also, $0<\alpha<180$, so
$$
0<55 k<360 \Rightarrow 0<k<6.6 \text {. }
$$

From equation (3), $k$ is an even number, thus, $k=2$ or 4. When $k=2$, $\alpha=55$; when $k=4$, $\alpha=110$.