6.6 All natural numbers from 1 to 50 inclusive are written on the board. Vasya chooses a pair of numbers on the board, the greatest common divisor of which is greater than one, and erases one of them. What is the smallest number of numbers Vasya can leave with such actions?
Solution. Evaluation. Note that we will not be able to choose the number 1 with any other number as a pair so that their GCD is greater than one, so one will definitely remain on the board. Also, note that the numbers $29,31,37,41,43$, and 47 will definitely remain on the board. They are prime and greater than half of 50. This means that all numbers divisible by them are not written on the board. Additionally, at least one number from the last pair chosen by Vasya must remain on the board. As we can see, we already have 8 numbers that must remain on the board.

Example. Let's show how to erase all odd numbers from the board, except those described in the evaluation. First, we will erase all odd composite numbers by pairing them with one of their prime divisors. Then, we will erase all odd primes less than 25 by pairing them with their doubled value (which will be even and therefore not erased by this point). Finally, we will erase all even numbers by pairing them with two. Note that indeed 8 numbers remain: $1,2,29,31,37,41,43$, and 47.

Answer. 8 numbers.