\begin{definition}[Definition:Classical Algorithm/Primitive Division]
The '''primitive''' operation for '''division''' which can be used in the classical algorithms is defined as follows.
Let $y$ be a $1$-digit integer.
Let $x$ be a $2$-digit integer $x_1 b + x_2$, where $x_1$ and $x_2$ are $1$-digit integers.
Suppose $x_1 \ge y$.
Then division of $x_1$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that:
{{begin-eqn}}
{{eqn | l = x_1
      | r = q \times y + r
}}
{{end-eqn}}
Suppose $x_1 < y$.
Then division of $x = x_1 b + x_2$ by $y$ gives a $1$-digit '''quotient''' $q$ and a $1$-digit '''remainder''' $r$, which is used as a '''carry''', such that:
{{begin-eqn}}
{{eqn | l = x
      | r = q \times y + r
}}
{{end-eqn}}
\end{definition}