jmc

An integer congruent to 1 (mod 9) can be written in the form $9n+1$ for some integer $n$. We want to count the number of integers $n$ such that $$1\le 9n+1\le 200.$$Subtracting 1 from all parts of the inequality, we get $0\le 9n\le 199$. Dividing by 9 we get $0\le n\le 22\frac{1}{9}$. There are $22-0+1=\boxed{23}$ values of $n$ corresponding to positive integers from 1 to 200 inclusive that are congruent to 1 (mod 9).