### Instruction: How many of the 200 smallest positive integers are congruent to 1 (mod 9)?### Response: To be congruent to 1 (mod 9) means to have a remainder of 1 when divided by 9.
The smallest positive integer that is congruent to 1 (mod 9) is 1 itself.
The next one is 1 + 9 = 10, and then 1 + 9 + 9 = 19, and so on.
I can see a pattern: the numbers congruent to 1 (mod 9) are those that can be written as 1 + 9k, where k is a non-negative integer.
I want to find the largest value of k such that 1 + 9k is still among the 200 smallest positive integers.
That means 1 + 9k ≤ 200, or 9k ≤ 199.
The largest integer k that satisfies this inequality is k = 22, because 9 * 22 = 198 and 9 * 23 = 207.