Question

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

Answer 1

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.