16. (12 points) There are $a$ matchsticks of the same length. When placed as shown in Figure 7, they can form $m$ squares; when placed as shown in Figure 8, they can form $2 n$ squares.
(1) Express $m$ in terms of $n$;
(2) When these $a$ matchsticks can also be arranged as shown in Figure 9, find the minimum value of $a$.


16. (1) The total number of matchsticks in Figure 7 is $3 m+1$, and the total number of matchsticks in Figure 8 is $5 n+2$.
Since the total number of matchsticks is the same, we have $3 m+1=5 n+2$.
Solving for $m$ gives $m=\frac{5 n+1}{3}$.
(2) Suppose there are $3 p$ squares in Figure 9, then the total number of matchsticks is $7 p+3$.
By the problem, we have $a=3 m+1=5 n+2=7 p+3$.
Thus, $p=\frac{3 m-2}{7}=\frac{5 n-1}{7}$.
Since $m$, $n$, and $p$ are all positive integers, when $m=17$, $n=10$, then $p=7$.
At this point, the minimum value of $a$ is
$$
a=3 \times 17+1=5 \times 10+2=7 \times 7+3=52 \text{. }
$$