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 º 2 mod 7. More generally m º p mod 7 means that m = p + 7k 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

If m º p mod 7 and n º q mod 7 prove that m + n º p + q mod 7 and mn º pq mod 7.