3. 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 number being divided.
Solution. Let the original number be $n$, and the "some other number" be $m$. The number $m$ is odd, because an even number has a divisor 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.

$n=4$ and $n=8$ work, corresponding to $m=9$ and $m=15$. Beyond that, nothing fits: the divisors $2,4,8$ turn into $3,5,9$, meaning $m$ has a proper divisor 15, but $n$ cannot have a divisor 14.