11.5. In one notebook, Vasya wrote down 11 natural numbers. In another notebook, Petya wrote down the greatest common divisors of each pair of numbers written in Vasya's notebook. It turned out that each number written in one of the two notebooks is also in the other notebook. What is the maximum number of different numbers that could have been written in Vasya's notebook?
Answer. 10.

Solution.

Answer. 10.

First, note that for any natural numbers $A \geq B$, the inequality $H C F(A ; B) \leq A$ holds, and equality is only achieved when $A=B$. Let $A \geq B$ be the two largest numbers in Vasya's notebook. Then in Petya's notebook, the number $A$ can only appear in the case when $A=B$; the HCF of all other pairs of numbers will be less than $A$. Therefore, there are no more than 10 different numbers in Vasya's notebook. If Vasya writes the numbers $2,4,8,16,32,64,128,256,512,1024,1024$ in his notebook, then in Petya's notebook, only the same numbers will be written.

Comment. Only an example with 10 different numbers - 2 points.

Only the estimate - 5 points.