# Modulo Arithmetic

Modulo arithmetic is concerned with the
division of whole numbers (or *integers*) into a
*quotient* and a *remainder*.

For example, 7 divided by 3 is 2 (the quotient), remainder 1. 12 divided by 3 is 4 remainder 0.

In this paper we refer to the quotient of two numbers, a and b as

aand the remainder asdivb

amodb

To use the above examples, 7 **div** 3 is 2, and
7 **mod** 3 is 1.

For a more complicated example 1996 **div** 19 is 105,
and 1996 **mod** 19 is 1 (because 1996 divided 19 is 105,
remainder 1).

A subtle point is that the remainder is always a positive number. So
-7 divided by 3 is -3, remainder 2 (rather than -2, remainder -1).
This is important for the date of Easter because we must often
calculate the remainder when dividing into a negative number. Since
in that case we are not interested in the quotient it is sufficient to
calculate the negative remainder and then simply add to it the number
we were dividing by. So -11 **mod** 3 may be calculated
as: -11 divided by 3 is -3, remainder -2, so the number required is
-2 + 3 = 1.

Simon Kershaw <simon@oremus.org>

10 February 2004