# Solution.

Let $n$ be such a number. In each pair of divisors, there is one of different parity, meaning there is an even one, so there is a divisor 2. Then there is a divisor 547. On the other hand, if $d$ is the greatest proper divisor, which is $n / 2$, then $d-545$ does not exceed $n / 3$. We get that

$$
\frac{n}{2}-545 \leqslant \frac{n}{3} \Longleftrightarrow \frac{n}{6} \leqslant 545 \Longleftrightarrow n \leqslant 6 \cdot 545
$$

Thus, $n$ is an even number, a multiple of 2.547, and does not exceed 6.545. The only candidates are $2 \cdot 547$ and $4 \cdot 547$. The first option works due to the simplicity of the number 547, the second, obviously, does not fit.
5. Find all natural numbers whose proper divisors can be divided into pairs such that in each pair the numbers differ by 545. A proper divisor of a natural number is a natural divisor other than 1 and the number itself.

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