How to estimate sums of residue classes mod $q$?

Question

Let gcd(a,q)=1. How do I estimate the sums $$\sum_{\substack{n\leq x\\ n\equiv a\operatorname{mod} q}}1.$$ Does it equal $\frac{x}{q}+O(1)$?

Answer 1

Assume $a \geq 0$. You are interested in all nonnegative integers $m$ such that $a + mq \leq x$. The number of such elements is $$\left\lfloor \frac{x - a}{q} \right\rfloor + 1$$