## Problem of the Day #42: Subtracting divisors
*April 30, 2011*

*Posted by Mitchell in : potd , trackback*

A number $n$ is written on a blackboard. In one turn, Albert may choose a divisor $d$ of the number $k$ written on the board and replace $k$ by $k – d$. He wishes to change the number to $1$ in the least possible number of moves; call this number $f(n)$.

For how many integers $n < 1000$ is $2^{f(n) – 1} \leq n$?

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