Question 164
We have an infinite list of boxes, numbered $1,2, \ldots$. Initially, every box contains the number $1$. At each step, we choose a number $a \in \mathbb{N}^*$ such that:
- either all boxes whose number is a multiple of $a$ contain $1$, in which case we replace these $1$s by $0$s,
- or all boxes whose number is a multiple of $a$ contain $0$, in which case we replace these $0$s by $1$s.
Show that, for every $n \geqslant 2$, one can arrange that all boxes numbered at most $n$ contain a $0$, except the first box, which contains a $1$.
- either all boxes whose number is a multiple of $a$ contain $1$, in which case we replace these $1$s by $0$s,
- or all boxes whose number is a multiple of $a$ contain $0$, in which case we replace these $0$s by $1$s.
Show that, for every $n \geqslant 2$, one can arrange that all boxes numbered at most $n$ contain a $0$, except the first box, which contains a $1$.