7.4. All natural numbers from 1 to 100 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. In order to erase a number, Vasya must find two numbers in whose prime factorization there will be at least one identical prime number. However, the number one (which has no prime numbers at all) and prime numbers greater than 50 (for which there is no second number that divides by it) cannot be erased. Prime numbers greater than 50 are 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 - a total of 10 numbers. Furthermore, when one of the numbers in a pair containing the same prime number is erased, the other remains on the board. Therefore, at the last erasure, there must be at least one number left that contains a prime divisor less than 50. In total, at least 12 numbers will remain on the board.

Example. Vasya can act as follows. For each prime number less than 50, there is also a doubled one on the board. Therefore, by choosing a prime $p$ in a pair with $2p$, Vasya can erase the prime numbers from 11 to 47, temporarily leaving $2, 3, 5,$ and 7. Each composite number up to 100 inclusive has a prime divisor less than 10, so Vasya can erase all composite numbers except 6, 10, and 14. Then Vasya can remove 3, 5, and 7 in the same way as the other prime numbers. Finally, using the number 2, remove 6, 10, and 14. In the end, the numbers $1, 2, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$ will remain on the board.

Answer. 12 numbers.