Example 1 The relationship between the sets $M=\{u \mid u=12 m+8 n+4 l$, $m, n, l \in \mathbf{Z}\}$ and $N=\{u \mid u=20 p+16 q+12 r$, $p, q, r \in \mathbf{Z}\}$ is ( ).
(A) $M=N$
(B) $M \nsubseteq N, N \nsubseteq M$
(C) $M \subset N$
(D) $M \supset N$
(1989, National High School Mathematics Competition)
Explanation: Starting from the expressions of elements in two sets.
Since $12 m+8 n+4 l=4(3 m+2 n+l)$,
$$
20 p+16 q+12 r=4(5 p+4 q+3 r) \text {, }
$$

and $(3,2,1)=1,(5,4,3)=1$, by Bézout's theorem, $3 m+2 n+l$ and $5 p+4 q+3 r$ can both represent all integers. Therefore, $M=N=\{k \mid k=4 l, l \in \mathbf{Z}\}$.
Hence, the answer is (A).