Game of Divisors

Two immensely intelligent players, A & B, engage in a game, the rules of which are as follows. For a natural number $N$, the board consists of numbers from $1$ to $N$. Each player takes turns to strike off a (new) number from the board. But, to make sure $N$ doesn't affect who wins, there is an added rule. Once you strike off a number, you also have to strike off all its divisors in that same chance, irrespective of whether any of those divisors were already marked. The player to strike off the last number on the board wins.

Can A construct a winning strategy?

Note: The question asks for the existence of the strategy, not to devise the strategy itself.