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 $$4 \times 2 = 8 = 1\left( 7 \right) + 1$$.