### 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**** **.