How many of the 1000 smallest positive integers are congruent to 1 modulo 9?

1 modulo 9 is the set of numbers as follows:

(0 (9) + 1), (1 (9) + 1, ...)

which is:

1, 10, 19

All we have to do is divide 1000 by 9, and round down.

1000/9 = 111.111 ...

There are 111 numbers.

Now let us check our work.

the set is as follows:

(1,10,19, ... (111) 9 + 1)

(1, 10, 19 ... 1000)

We are within 1000.

Therefore, 111 of the 1000 smallest positive integers are congruent to 1 modulo 9.