1. Milla and Zhena came up with a number each and wrote down all the natural divisors of their numbers on the board. Milla wrote 10 numbers, Zhena - 9, and the number 6 was written twice. How many different numbers are on the board?
Solution. Since the number 6 is written twice, both original numbers (denote them as a and b) are divisible by 6.

If Vera's number has 10 divisors, then its factorization is either $p^{9}$ or $p^{1} \cdot q^{4}$ (where p and q are some prime numbers); the first is impossible since it is divisible by 6. Valya's number has 9 divisors, so its factorization is either $\mathrm{s}^{8}$ or $\mathrm{s}^{2} \mathrm{t}^{2}$; again, only the second is possible. In this case, the numbers p and q are 2 and 3 in some order, and the numbers $\mathrm{s}$ and $\mathrm{t}$ are also 2 and 3. It is easy to see that the GCD of such numbers is either $2 \cdot 3^{2}=18$ or $3 \cdot 2^{2}=12$, and in any case has $2 \cdot 3=6$ divisors. Therefore, among the listed numbers, exactly 6 are repeated, and the number of distinct numbers is $10+9-6=13$.

## Criteria.

A complete solution should include: (1) an indication of the form of the GCD factorization, with proof; (2) the calculation of the divisors of this GCD and the derivation of the answer, as in problem 7.1. Providing an example confirming the existence of such numbers is not required.

Errors. A person counts only the numbers that are written once (and gets the answer 7) — minus one point.

If the GCD factorization is correctly found but there are 1-2 errors in counting its divisors — deduct $1-2$ points.

Progress.

If the form of the GCD is correctly and justifiedly found, but then something goes wrong — 3 points can be given.

Reasonable attempts to determine the factorization of each of the original numbers — 1 point.

Incorrect but partially reasonable solutions.

"The numbers $1,2,3,6$ are written twice, so the number of different numbers is $10+9-4=15$ " — 1 point.

An example of suitable numbers is provided and the answer is found for it — 2 points.