Question 12
For positive integers $n,$ let $\tau(n)$ denote the number of positive integer divisors of $n,$ including $1$ and $n.$ For example, $\tau(1) = 1$ and $\tau(6) = 4.$ Define $S(n)$ by $$
S(n) = \tau(1) + \tau(2) + \cdots + \tau(n).
$$ Let $a$ denote the number of positive integers $n \le 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \le 2005$ with $S(n)$ even. Find $|a - b|.$
S(n) = \tau(1) + \tau(2) + \cdots + \tau(n).
$$ Let $a$ denote the number of positive integers $n \le 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \le 2005$ with $S(n)$ even. Find $|a - b|.$