5. An Indian engineer thought of a natural number, listed all its proper natural divisors, and then increased each of the listed numbers by 1. It turned out that the new numbers are all the proper divisors of some other natural number. What number could the engineer have thought of? Provide all options and prove that there are no others. Recall that a divisor is called proper if it is not equal to 1 and does not coincide with the dividend.
Solution. Answer: 4 or 8.

Let the original number be $n$, and "some other number" be $m$. The number $m$ is odd because an even number has a divisor of 2, which could not have resulted. Therefore, all its divisors are odd. This means all divisors of $n$ are even. Thus, $n$ is a power of two. The options $n=4$ and $n=8$ work, corresponding to $m=9$ and $m=15$. Larger powers of two do not work: the divisors $2, 4, 8$ after the engineer's actions turn into $3, 5, 9$, meaning $m$ has a proper divisor of 15, but $n$ cannot have a divisor of 14.