I Positive or negative remainder

Usually we consider entire equivalence classes in such cases: Every single element of ##\{\ldots -13, -8, -3, 2, 7 , 12, \ldots\}## belongs to the same remainder of a division by ##5##. We then define all five possible classes
##\{\ldots -15, -10, -5, 0, 5, 10, \ldots\}##
##\{\ldots -14, -9, -4, 1, 6, 11, \ldots\}##
##\{\ldots -13, -8, -3, 2, 7, 12, \ldots\}##
##\{\ldots -12, -7, -2, 3, 8, 13, \ldots\}##
##\{\ldots -11, -6, -1, 4, 9, 14, \ldots\}##
as elements of a new set with five elements ##\{ \; \{\ldots -15, -10, -5, 0, 5, 10, \ldots\}\, , \, \{\ldots -14, -9, -4, 1, 6, 11, \ldots\}\, , \, \ldots \}##.

This notation is a bit nasty to handle, so we choose one representative out of every set. E.g. ##\{[-15],[-9],[12],[3],[-1]\}## could be chosen, but this is still a bit messy to do calculations with. So the most convenient representation is ##\{[0],[1],[2],[3],[4]\}## with the non-negative remainders smaller than ##5##. However, this is only a convention. ##-3## is a remainder, too, belonging to the class ##[2]##. So the answer to your questions is: The statement is true, as all integers are remainders.

 

The remainder is usually required to be between 0 and N-1 inclusive. 23 and -2 (not -3) are in the same equivalence class. This can also be written as 23 = -2 mod 5.