My professor has this one line in his homework that has me completely tripping:
"Note that if x (mod y)= z (mod y), then x−z = ky for some integer k. We can use this property to extend mod to negative numbers as well, i.e., −4 (mod 6) = 2, −1024912489 (mod 10) = 1. "
What does he mean by "extend"? I don't see how this property extends anything, and I don't understand what it has to do with negative mod.
Help?
Your professor is referring to the fact that if $$x-z=ky$$ then you can say $$x\equiv z \ \mod y $$ without worrying whether $x$ or $z$ are negative.