Clarifying meaning of "mod"

Question

I'm having trouble understanding exactly how $mod$ works. I know it's remainder, but I guess my problem is in how to write it out.

For example, $\frac{1}{3} = 0$ remainder $1$.

Would I write $1 = 0(mod 3)$, $1 = 1 (mod 3)$...?

Edit: So, the base goes in the (mod) brackets, the number being divided is on the left, and the remainder is on the right?

Answer 1

Based of its definition, Modulo operation is to find the "REMAINDER AFTER DIVISION" Since it's target is the remainder, "You mod something" = "the remainder".

You mod something is between something which in turns produce that remainder. Obviously its division. Now who divided by who gets that remainder? Those in the mod would by definition be equivalent to what we can call the BASE.

How would you figure out for the rest? :D