Multiplication modulo

Now here we are going to define another new type of multiplication, which is known as “multiplication modulo p.” It can be written as a{ \times _p}b, where a and b are any integers and p is a fixed positive integer.


a{ \times _p}b = r,\,for\,0 \leqslant r \leqslant p


Here r is the least non-negative remainder when ab, i.e. the ordinary product of a and b, is divided by p. For example 4{ \times _7}2 = 1, since 4 \times 2 = 8 = 1\left( 7 \right) + 1.