Dividing Whole Numbers (Long Division)

In some sense dividing whole numbers is the inverse of multiplying whole numbers.

Result of dividing number `a` by number `b` is number `c=a/b` such that `a=b times c`.

For example, since `12=3 times 4` then `4=12/3`.

Division is denoted by symbol a/b (shown as `(a)/(b)`). We will use this symbol, but also common symbols are `-:` and `:`

Thus, `12/4`, `12-:4` and `12:4` are equivalent.

To understand division, think about division as a process when we try to find out how many times a number (divisor) is contained in another number (dividend).

Result of the division is called quotient.

For example, since `12=3+3+3+3` or `12=3 times 4`, we conclude that number 3 is contained in 12 four times, thus `12/3=4`. 3 is divisor, 12 is dividend, 4 is quotient.

Also, there is another important fact.

We can't divide by zero: `a/0` is undefined.

Indeed, we can't calculate how many times zero is contained in number `a`.

For any number `a` we have that `0/a=0` (number `a` is contained 0 times in 0).

For example, `0/10=0`.

Indeed, number 10 is contained 0 times in 0.

It is pretty easy to determine how many times small number is contained in another number, but it becomes hard to work with big numbers. For example, can you say how many times is number 12 contained in 2184? Yes, it is hard.

Below you will understand how to find result of division of any number by any number, but, for now, let's start from simple example.

Example 1. Find `86/2`.

Write in special form:

$$$\begin{array}{r}\\2\hspace{1pt})\overline{\hspace{1pt}86}\\\end{array}$$$

First let's divide 8 by 2. How many 2s are in 8? 2+2+2+2=8. There are four 2s, so `8/2=4`.

Write down 4.

Multiply 4 by 2. Result is 8. Write it down.

$$$\begin{array}{r}\color{green}{4}\phantom{6}\\\color{green}{2}\hspace{1pt})\overline{\hspace{1pt}86}\\-\underline{\color{green}{8}}\phantom{6}\\\end{array}$$$

Now subtract 8 from 8. Result is 0.

$$$\begin{array}{r}4\phantom{6}\\2\hspace{1pt})\overline{\hspace{1pt}\color{purple}{8}6}\\-\underline{\color{purple}{8}}\phantom{6}\\\color{purple}{0}\phantom{6}\end{array}$$$

Drag 6 down.

$$$\begin{array}{r}4\phantom{6}\\ 2\hspace{1pt})\overline{\hspace{1pt}8\color{red}{6}}\\-\underline{8}\phantom{6}\\0\color{red}{6}\end{array}$$$

So, what have we done?

We've first done division, then multiplication, then subtraction.

Let's proceed in the same way.

Divide 6 by 2. Result is 3.

Now, multiply 2 and 3. Result is 6.

$$$\begin{array}{r}4\color{blue}{3}\\ \color{blue}{2}\hspace{1pt})\overline{\hspace{1pt}86}\\-\underline{8}\phantom{6}\\06\\-\underline{\phantom{0}\color{blue}{6}}\end{array}$$$

Now, subtract 6 from 6. Result is 0.

$$$\begin{array}{r}43\\ 2\hspace{1pt})\overline{\hspace{1pt}86}\\-\underline{8}\phantom{6}\\0\color{purple}{6}\\-\underline{\phantom{0}\color{purple}{6}}\\\color{cyan}{0}\end{array}$$$

We are done because we obtained zero and there are no numbers to divide.

So, `86/2=43`.

Let's do another example.

Example 2. Find `72/3`.

Write in special form:

$$$\begin{array}{r} 3\hspace{1pt})\overline{\hspace{1pt}72}\\\end{array}$$$

How many 3s are in 7? 3+3+1=7. There are two 3s and something extra.

Write down 2.

Now, multiply 3 by 2. Result is 6. Write it down.

$$$\begin{array}{r}\color{green}{2}\phantom{2}\\\color{green}{3}\hspace{1pt})\overline{\hspace{1pt}72}\\-\underline{\color{green}{6}}\phantom{0}\\ \end{array}$$$

Now subtract 6 from 7. Result is 1.

$$$\begin{array}{r}2\phantom{2}\\3\hspace{1pt})\overline{\hspace{1pt}\color{purple}{7}2}\\-\underline{\color{purple}{6}}\phantom{2}\\\color{purple}{1}\phantom{2}\end{array}$$$

Drag 2 down.

$$$\begin{array}{r}2\phantom{2}\\3\hspace{1pt})\overline{\hspace{1pt}7\color{red}{2}}\\-\underline{6}\phantom{2}\\1\color{red}{2}\end{array}$$$

Next, continue with 12.

How many 3s are in 12? 3+3+3+3=12. There are four 3s.

Now, multiply 4 and 3. Result is 12.

$$$\begin{array}{r}2\color{blue}{4}\\\color{blue}{3}\hspace{1pt})\overline{\hspace{1pt}72}\\-\underline{6}\phantom{2}\\12\\-\underline{\color{blue}{12}}\end{array}$$$

Subtract 12 from 12. Result is 0.

$$$\begin{array}{r}24\\3\hspace{1pt})\overline{\hspace{1pt}72}\\-\underline{6}\phantom{2}\\\color{purple}{12}\\-\underline{\color{purple}{12}}\\\color{cyan}{0}\end{array}$$$

We are done, because we obtained zero and there are no numbers to divide.

So, `72/3=24`.

Finally, let's do a harder example.

Example 3. Find `2184/12`.

$$$\begin{array}{r} 12\hspace{1pt})\overline{\hspace{1pt}2184}\\ \end{array}$$$

How many 12s are in 2? Zero! 12 is greater than 2.

So, we just add next digit (take 21 instead of 2): how many 12's are in 21? 12+9=21. There is one.

Write down 1.

Multiply 12 by 1. Result is 12. Write it down.

$$$\begin{array}{r}\color{green}{1}\phantom{184}\\\color{green}{12}\hspace{1pt})\overline{\hspace{1pt}2184}\\-\underline{\color{green}{12}}\phantom{84}\\ \end{array}$$$

Subtract 12 from 21. Result is 9.

$$$\begin{array}{r}1\phantom{184}\\12\hspace{1pt})\overline{\hspace{1pt}\color{purple}{21}84}\\ -\underline{\color{purple}{12}}\phantom{84}\\\color{purple}{9}\phantom{84}\end{array}$$$

Drag 8 down.

$$$\begin{array}{r}1\phantom{184}\\12\hspace{1pt})\overline{\hspace{1pt}21\color{red}{8}4}\\- \underline{12}\phantom{84}\\9\color{red}{8}\phantom{4}\end{array}$$$

Continue with 98.

How many 12s are there in 98? 12+12+12+12+12+12+12+12+2=98. There are eight 12s.

Multiply 12 and 8. Result is 96.

$$$\begin{array}{r}1\color{blue}{8}\phantom{84}\\\color{blue}{12}\hspace{1pt})\overline{\hspace{1pt}2184}\\-\underline{12}\phantom{84}\\98\phantom{4}\\-\underline{\color{blue}{96}}\phantom{4}\end{array}$$$

Subtract 98 from 96. Result is 2.

$$$\begin{array}{r}18\phantom{84}\\12\hspace{1pt})\overline{\hspace{1pt}2184}\\-\underline{12}\phantom{84}\\\color{green}{98}\phantom{4}\\-\underline{\color{green}{96}}\phantom{4}\\\color{green}{2}\phantom{4}\end{array}$$$

Drag down 4.

$$$\begin{array}{r}18\phantom{84}\\12\hspace{1pt})\overline{\hspace{1pt}218\color{red}{4}}\\-\underline{12}\phantom{84}\\98\phantom{4}\\-\underline{96}\phantom{4}\\2\color{red}{4}\end{array}$$$

Finally, determine how many 12s in 24? 12+12=24. There are two 12s.

Multiply 12 by 2. Result is 24.

Subtract 24 from 24. Result is 0.

$$$\begin{array}{r}18\color{purple}{2}\phantom{4}\\\color{purple}{12}\hspace{1pt})\overline{\hspace{1pt}2184}\\-\underline{12}\phantom{84}\\98\phantom{4}\\-\underline{96}\phantom{4}\\24\\-\underline{\color{purple}{24}}\\\color{cyan}{0}\end{array}$$$

We are done because we obtained zero and there are no numbers to divide.

So, `2184/12=182` .

Now, it is your turn. Take pen and paper and solve the following problems:

Exercise 1. Find `96/3`.

Answer: 32.

Exercise 2. Find `64/4`.

Answer: 16.

Exercise 3. Find `125/5`.

Answer: 25.

Exercise 4. Find `1632/24`.

Answer: 68.

Exercise 5. Find `59004/447`.

Answer: 132.