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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# 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.