Example 5. Five monkeys found Wang Yiguan's peaches. How can they be fairly divided when one monkey secretly gets up, eats one peach, and the remaining peaches are exactly divided into five equal parts, after which it hides its share and goes back to sleep. The second monkey gets up, also eats one peach, and the remaining peaches are again exactly divided into five equal parts, after which it also takes its share and goes back to sleep. The third, fourth, and fifth monkeys all do the same. How many peaches were there at the very beginning?
This problem has many solutions, for example, it can first be transformed into an indeterminate equation, and then its positive integer solutions can be sought. However, there is a simpler solution: Imagine that when Sun Wukong, the Great Sage Equal to Heaven, found his subordinates facing such a difficult problem, he had a kind heart and pulled out four hairs from his sleeve to turn into four monkeys, who added to a pile of peaches while the Monkey King was sleeping. Thus, when the first monkey came, it could exactly divide the pile of peaches into five equal parts. Suppose the part it took did not contain any peaches transformed from the hairs, and the quantity it obtained was the same as what the others would get (five equal parts). This scenario can be deduced further. Because the integer 4, plus the four hair-transformed monkeys, the number of peaches in the pile can be divided by 5 five times in succession, which means: the original pile of peaches had a minimum number of
$$
5^{5}-4=3121.
$$

The mathematical puzzle in the above example 4 can actually serve as a math competition problem for elementary school students. We can see closely that such problems can also fascinate the greatest scholars. This fully demonstrates that good mathematical puzzles are the wonders in the garden of wisdom that all those who are diligent in thinking can enjoy.