Example 14 Two gears, A and B, are meshed together. Gear A has 437 teeth, and gear B has 323 teeth. After a certain tooth on A and a certain tooth on B come into contact, how many minimum revolutions will each have to make before they contact each other again?

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To find the minimum number of weeks each wheel must turn, we first need to determine the number of teeth each wheel (A and B) must pass. To do this, we need to find the least common multiple (LCM) of the number of teeth on wheel A (437) and wheel B (323). Since
\begin{tabular}{|c|c|c|}
\hline & \multicolumn{2}{|r|}{323} \\
\hline & 323 & 228 \\
\hline & 114 & 95 \\
\hline & 95 & 95 \\
\hline & 19 & 0 \\
\hline
\end{tabular}

Thus, $(437,323)=19$. By Lemma 10, we have
$$\{437,323\}=\frac{437 \times 323}{19}=7429$$

Therefore, the number of weeks wheel A turns: $7429 \div 437=17$ weeks, and the number of weeks wheel B turns: $7429 \div 323=23$ weeks.