How to divide using addition or subtraction We can multiply $a$ and $n$ by adding $a$ a total of $n$ times. 
$$ n \times a = a + a + a + \cdots +a$$
Can we define division similarly using only addition or subtraction?
 A: You can also use additions. One should use results from intermediate calculations to speed up.
Let us divide 63 by 12.
$$
\begin{split}
12+12=24,&\qquad\textrm{count }1+1=2\\
24+24=48,&\qquad\textrm{count }2+2=4\\
48+24=72,&\qquad\textrm{count }4+2=6\textrm{ (exceeded 63)}\\
48+12=60,&\qquad\textrm{count }4+1=5\textrm{ (so we try adding less)}\\
63-60=3,&\qquad\textrm{(calculation of the remainder)}\\
\end{split}
$$
A: To divide $60$ by $12$ using subtraction:
$$\begin{align*}
&60-12=48\qquad\text{count }1\\
&48-12=36\qquad\text{count }2\\
&36-12=24\qquad\text{count }3\\
&24-12=12\qquad\text{count }4\\
&12-12=0\qquad\;\text{ count }5\;.
\end{align*}$$
Thus, $60\div 12=5$.
You can even handle remainders:
$$\begin{align*}
&64-12=52\qquad\text{count }1\\
&52-12=40\qquad\text{count }2\\
&40-12=28\qquad\text{count }3\\
&28-12=16\qquad\text{count }4\\
&16-12=4\qquad\;\text{ count }5\;.
\end{align*}$$
$4<12$, so $64\div 12$ is $5$ with a remainder of $4$.
A: If $n$ is divisible by $b$ ($\frac{n}{b}$ is a whole number), then keep doing $n - b - b - b - b - b - \cdots - b$ until the value of that is $0$. The number of times you subtract $b$ is the answer. For example, $\frac{20}{4} \rightarrow 20 - 4 - 4 - 4 - 4 - 4$. We subtracted '$4$' five times, so the answer is $5$.
A: You can define division as repeated subtraction:$${72\over 9}=72-9-9-9-9-9-9-9-9$$Subtracting by $9$ eight times is the same as subtracting by $72$ since $9\cdot8=72$.  So, the answer is $8$.  Also, this is why ${n\over a}=n-a-a-a-a\cdots$ for whatever whole number $a$ is other than zero.
If you have a remainder, then you just do this:$${13\over 2}=13-2-2-2-2-2-2-1$$as you just saw, subtracting by $2$ six times is the same as subtracting by $12$ since $2\cdot6=12$, but there's a remainder of $1$ being sutracted, so it's the same as subtracting by $13$ since $2\cdot6+1=13$, so the answer is $6$ R$1$ or $6.5$.
