How do I solve for negative modular arithmetic?

Question

Here is an example: -45 mod 13 = 7, but how?

Answer 1

a mod b = (a + nb) mod b

so, -45 mod 13 = mod (-45 + 52) mod 13 = 7 mod 13 = 7

Just add a big enough multiple of the modulus so you have a positive number.