Question

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

Answer 1

Positive integers congruent to 1 modulo 9 are 1,10,19,28, 37, 46, 55,...,991, 1000
This is an arithmetic series where
`a_1=1,a_n=1000,d=9,n=?`


`a_n=a_1+(n-1)d \\1000=1+(n-1)9 \\999=9(n-1) \\111=n-1 \\n=112`

Total number of positive integers congruent to 1 modulo 9 is 112.