Here we have two different types of arithmetic system, on one hand 'modulus' (or clock) arithmetic and on the other hand, place value systems (which use different number bases). Unlike modulus arithmetic, where many numbers are represented by the same symbol, in base arithmetic each distinct number has its own distinct representation different from that of any other number.

The table compares counting in the two systems, using arithmetic modulo 7 and numbers in base 7 as an example. The units digit of a number in base 7 is the same as the equivalent number in arithmetic modulo 7.

Clock arith. 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3
Base arith. 1 2 3 4 5 6 10 11 12 13 14 15 16 20 21 22 23
In base 7, numbers have the units digits 0, 1, 2, 3, 4, 5 and 6. In arithmetic modulo 7, any integer m is represented by (or said to be congruent to) one of the numbers 0, 1, 2, 3, 4, 5, or 6 which is given by the remainder when m is divided by 7.

For example the decimal number 33 is written as 45 in the base 7 because 33 = 4*7 + 5 and 33 is congruent to the number 5 in arithmetic modulo 7. The decimal number 80 is written 143 in the base 7 system because 80 = 11*7 + 3 = 1*49 + 4*7 + 3 and 80 is congruent to the number 3 in arithmetic modulo 7.

Vassil Vassilev, age 15, Lawnswood High School, Leeds gave the following proof that, in modulus arithmetic, when we add or multiply two numbers, we get the same result as when we first find numbers to which they are congruent and then add or multiply those numbers.

If m=p+7r and n=q+7s (where m,n,p,q,r,s are integers) we can say that m and n are congruent to p and q modulo 7 respectively and we write: m≡pmod7 and n≡qmod7.
m+n =p+7r+q+7s m+n =p+q+7(r+s)wherer+sis an integer. m+n ≡(p+q)mod 7.


mn =(p+7r)(q+7s) mn =pq+7ps+7rq+49rs =pq+7(ps+rq+7rs)where (ps+rq+7rs)is an integer. mn ≡(pq)mod 7.