## Identification numbers with check digit algorithms

### Modular multiplication

(Modular Arithmetic)

In standard arithmetic, $$5 \times 10$$ is equal to $$50$$, but in Modular Arithmetic it is equal to $$2$$, since this is the remainder of the divison of $$50$$ by $$12$$. This is usually expressed as $5 \times 10=2\,(\mbox{mod }12).$

What about modulus $$9$$ instead of modulus $$12$$? Likewise, since $$50 = 5 \times 9 + 5$$, we say that $5 \times 10=5\,(\mbox{mod }9).$

In order to check your ability with modular multiplication, see this applet.

Modular multiplication has the following properties:

• It is commutative: $$a \times b$$ is equal to $$b \times a$$ for every $$a$$ and $$b$$;
• It has an identity element (precisely the number 1, since $$a \times 1 = a$$ for every $$a$$)
• Every element (different from 0) has an inverse only when the modulus is a prime $$p$$. In this case, the inverse of $$a$$ is the number $$b$$ such that $$b \times a = 1$$.

Check these and other properties of multiplication in the following "couloured" Multiplication Table .