10. Find the sum of the greatest odd factor of each of the numbers $2007,2008, \ldots, 4012$.求 $2007 、 2008 、 \cdots 、 4012$ 各數的最大奇因數之和。
10. 4024036
10. For a positive integer $n$, let $p(n)$ denote its greatest odd divisor. Then $n=2^{k} p(n)$ for some nonnegative integer $k$. Hence if $p(r)=p(s)$ for $r \neq s$, then one of $r$ and $s$ is at least twice the other. Because no number from $2007,2008, \ldots, 4012$ is at least twice another, $p(2007)$, $p(2008), \ldots, \quad p(4012)$ are 2006 distinct odd positive integers. Note further that each of them must be one of the 2006 odd numbers $1,3,5, \ldots, 4011$. It follows that they are precisely 1,3 , $5, \ldots, 4011$ up to permutation, and so the answer is
$$
1+3+5+\cdots+4011=2006^{2}=4024036 \text {. }
$$