Circle with numbers 0 to 6.

If today is Monday we know that in 702 days time (that is in 100 weeks and 2 days time) it will be Wednesday. This is an example of "clock" or modulus arithmetic. We say

702

is congruent to

2

modulo

7

and write

702 \equiv 2 mod 7

. More generally

m \equiv p mod 7

means that

m = p + 7 k

where

p

is an integer between

0

and

6

inclusive and

k

is an integer. "

Investigate and explain the link between how numbers are recorded in modulus arithmetic and the units digit when the number is written in a different number base. Here is an example of a few randomly chosen numbers written as numbers modulo 7 and written in base 7). Investigate other number bases.

Some numbers in base ten	1	2	9	11	26	33	41	67	80	100
The same numbers modulo 7	1	2	2	4	5	5	6	4	3	2
The same numbers in base 7	1	2	12	14	35	45	56	124	143	202

m \equiv p mod 7

and

n \equiv q mod 7

prove that

m + n \equiv p + q mod 7

and

mn \equiv pq mod 7