6. Vasya and Petya thought of two different numbers, and each of them turned out to have as many prime divisors as composite ones. Can Vasya's and Petya's numbers have common divisors greater than 1?
Solution. Let the number $n$ have more than two prime divisors, counting multiplicity. Then for each prime $p$ dividing $n$, the number $n / p$ is composite (and all these numbers are distinct). Also, $n$ divides itself, so there are more composite divisors than prime ones. Therefore, there cannot be more than two prime factors. It is easy to see that there cannot be just one, so there must be two, i.e., $n = pq$, where $p, q$ are primes. But the only composite divisor here is $pq$. Thus, $p = q$, meaning $n = p^2$. Clearly, any two different numbers of this form do not have common divisors (other than 1), so the answer is no.