Modulus with negative remainder

Question

It has previously been asked about -a mod b, and, summarizing my gain:

-1000 % 9
-1000/9 = -111.111....
9*112-1000 = 8

This is the method I've found to be effective.
1. What is the easiest way to solve this without calculator?
2. Is there a better way to reach this answers with calculator?

Answer 1

There are lots of tricks. For example, $$ -1000\equiv-\left(10\right)^{3}\equiv-\left(1\right)^{3}\equiv-1\equiv8\mod{9} $$ Since $$ 10\equiv1\mod{9}. $$ I think you should start by reading the Wikipedia page on modular arithmetic in detail.