Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.
Problem
Source: 2021 Greek Junior MO p2 (served as Greek JBMO TST p2 since the latter didn't take place)
Tags: game, game strategy, combinatorics, winning strategy, prime divisor