# almanac

# The Date of Easter

## 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

*a* **div** *b*

and the remainder as
*a* **mod** *b*

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 by 19 is 105,
remainder 1).

### Negative numbers

A subtle point is that the remainder is always a positive number in the
range 0 to *b*−1, where *b* is the divisor (or modulus). 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, and some computer
implementations of the modulus function do not behave in this way with
negative numbers. Since in that case we are not interested in the quotient
it is sufficient to check whether a negative remainder has been calculated
and, if so, then 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.

### Using a different range

Although the standard modulo operation gives a result in the range 0
to *b*−1, it is also possible to offset the result to a
different range. In general, if the offset is *c* then the
result will lie in the range *c*
to *c*+*b*−1. Although there is no standard
notation, this can be written as

a **mod**_{c} b

In mathematical terms values calculated with a different offset are *congruent* to one another.

This offset notation is a convenience and can easily be rewritten using a standard **mod** operator:

a **mod**_{c} b
≡
(a − c) **mod** b + c

### About this page

This page is part of the oremus Almanac by Simon Kershaw.
almanac.oremus.org/easter/computus