10.4. All divisors of a certain natural number have been listed, except for one and the number itself. Some two numbers in this list differ by six times. By how many times do the two largest numbers in this list differ?
Answer: one and a half times.

Solution. Let among the divisors of the number $N$ there be numbers $a$ and $6a$, then $N$ is divisible by $6a$. Therefore, $N$ is divisible by 2 and by 3, which means 2 and 3 are the two smallest numbers in the list. Then the two largest numbers in the list are $\frac{N}{3}$ and $\frac{N}{2}$. Their ratio is: $\frac{N}{2} : \frac{N}{3} = \frac{3}{2}$.

Grading criteria.

“+" A complete and well-reasoned solution is provided

"士" A generally correct solution is provided, containing minor gaps or inaccuracies “Ғ” The correct answer is obtained, based on the fact that 2 and 3 are the smallest numbers in the list of divisors, but their presence in this list is not proven

“〒” The correct answer is obtained, based on the consideration of specific examples

“-" Only the answer is provided

"-" An incorrect solution is provided or it is absent