4. A divisor of a natural number is called proper if it is greater than 1 and less than the number itself. A natural number is called elegant if it has at least two proper divisors and is divisible by the difference of any two of them. Find all elegant numbers.
Answer: 6, 8, 12. Solution. It is easy to verify that the numbers 6, 8, and 12 are elegant. Let $\mathrm{N}$ be an arbitrary elegant number. An odd number has only odd divisors, the differences between divisors are even, and an odd number cannot be divided by an even number. Therefore, $N=2 n$. But then $2 n$ must be divisible by $n-2$, which means that $4=2 n-2(n-2)$ must also be divisible by $n-2$. Thus, $n-2$ equals $4, 2, 1$, or -1. The case $n=1$ is excluded, since the number 2 does not have proper divisors, and the other three cases give the three elegant numbers specified in the answer.