3-ча 1. The sum of the reciprocals of three positive integers is equal to 1. What are these numbers? Find all solutions.
Solve 1. Answer: \((2,4,4),(2,3,6)\) or \((3,3,3)\).

Let \(x \leqslant y \leqslant z\) be natural numbers, and \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\). Then \(x1000^{10}\), so the number \(2^{100}\) has at least 31 digits. On the other hand,

\[
\frac{1024^{10}}{1000^{10}}<\left(\frac{1025}{1000}\right)^{10}=\left(\frac{41}{40}\right)^{10}<\frac{41}{40} \cdot \frac{40}{39} \cdot \frac{39}{38} \cdots \frac{32}{31}=\frac{41}{31}<10
\]

Thus, \(2^{100}=(1024)^{10}<10 \cdot 1000^{10}\), so the number \(2^{100}\) has fewer than 32 digits.