Division with Remainder

When we divided whole numbers, we gave such examples that result is whole number.

But in general it is not true.

What if we try to divide 11 by 4?

There are two 4s in 11 and something extra: `11=4+4+3` or `11=4*2+3`.

This is called division with remainder. It occurs when result of division is not whole number.

In general if `m` is dividend, `n` is divisor, `q` is quotient and `r` is remainder then `color(green)(m=n*q+r)`. Note, that `n` should be greater than `r`. All numbers should be whole numbers.

In the above example we could write `11=4*1+7`. Although it is correct, but here `4<7`, so remainder is found incorrectly.

When remainder equals 0 then result of division is whole number.

For example, since `12/3=4` we can write that `12=4*3+0` or simply `12=4*3`.

In general, remainder can be found using the same technique, which we saw when divided whole numbers.

Let's go through a couple of examples.

Example 1. Find remainder after division of 87 by 2.

Write in special form:

$$$\begin{array}{r} 2\hspace{1pt})\overline{\hspace{1pt}87}\\ \end{array}$$$

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

$$$\begin{array}{r}\color{green}{4}\phantom{7}\\\color{green}{2}\hspace{1pt})\overline{\hspace{1pt}87}\\-\underline{\color{green}{8}}\phantom{7}\\ \end{array}$$$

Now subtract 8 from 8. Result is 0.

$$$\begin{array}{r}4\phantom{7}\\2\hspace{1pt})\overline{\hspace{1pt}\color{purple}{8}7}\\-\underline{\color{purple}{8}}\phantom{7}\\\color{purple}{0}\phantom{7}\end{array}$$$

Drag 7 down.

$$$\begin{array}{r}4\phantom{7}\\2\hspace{1pt})\overline{\hspace{1pt}8\color{red}{7}}\\-\underline{8}\phantom{7}\\0\color{red}{7}\end{array}$$$

So, what have we done?

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

Let's proceed in the same way.

Divide 7 by 2. There are three 2s and additional one: `7=2*3+1`.

Now, multiply 2 and 3. Result is 6.

$$$\begin{array}{r}4\color{blue}{3}\\\color{blue}{2}\hspace{1pt})\overline{\hspace{1pt}87}\\-\underline{8}\phantom{7}\\07\\-\underline{\phantom{0}\color{blue}{6}}\end{array}$$$

Now, subtract 7 from 6. Result is 1.

$$$\begin{array}{r}43\\ 2\hspace{1pt})\overline{\hspace{1pt}87}\\-\underline{8}\phantom{7}\\0\color{purple}{7}\\-\underline{\phantom{0}\color{purple}{6}}\\\color{cyan}{1}\end{array}$$$

We are done, because `1<2`, i.e. remainder is less than divisor.

So, remainder is 1:`87=2*43+1` .

Let's do another example.

Example 2. Find remainder after division of 74 by 3.

Write in special form:

$$$\begin{array}{r}3\hspace{1pt})\overline{\hspace{1pt}74}\\ \end{array}$$$

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

Write down 2.

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

$$$\begin{array}{r}\color{green}{2}\phantom{4}\\ \color{green}{3}\hspace{1pt})\overline{\hspace{1pt}74}\\-\underline{\color{green}{6}}\phantom{4}\\ \end{array}$$$

Now subtract 6 from 7. Result is 1.

$$$\begin{array}{r}2\phantom{4}\\3\hspace{1pt})\overline{\hspace{1pt}\color{purple}{7}4}\\-\underline{\color{purple}{6}}\phantom{4}\\\color{purple}{1}\phantom{4}\end{array}$$$

Drag 4 down.

$$$\begin{array}{r}2\phantom{4}\\ 3\hspace{1pt})\overline{\hspace{1pt}7\color{red}{4}}\\-\underline{6}\phantom{4}\\1\color{red}{4}\end{array}$$$

Next, continue with 14.

How many 3s are in 14? Four 3s and something extra: 14=3+3+3+3+2. There are four 3s.

Now, multiply 3 and 4. Result is 12.

$$$\begin{array}{r}2\color{blue}{4}\\\color{blue}{3}\hspace{1pt})\overline{\hspace{1pt}74}\\-\underline{6}\phantom{4}\\14\\-\underline{\color{blue}{12}}\end{array}$$$

Subtract 14 from 12. Result is 2.

$$$\begin{array}{r}24\\3\hspace{1pt})\overline{\hspace{1pt}74}\\-\underline{6}\phantom{4}\\\color{purple}{14}\\-\underline{\color{purple}{12}}\\\color{cyan}{2}\end{array}$$$

We are done, because we number that is less than 3 (divisor).

So, remainder is 2,`72=3*24+2` .

Finally, let's work through a slightly harder example.

Example 3. Find remainder after division of 2194 by 12.

$$$\begin{array}{r} 12\hspace{1pt})\overline{\hspace{1pt}2194}\\ \end{array}$$$

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 12s are in 21? 12+9=21. There is one.

Write down 1.

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

$$$\begin{array}{r}\color{green}{1}\phantom{94}\\\color{green}{12}\hspace{1pt})\overline{\hspace{1pt}2194}\\-\underline{\color{green}{12}}\phantom{94}\\ \end{array}$$$

Subtract 12 from 21. Result is 9.

$$$\begin{array}{r}1\phantom{94}\\ 12\hspace{1pt})\overline{\hspace{1pt}\color{purple}{21}94}\\-\underline{\color{purple}{12}}\phantom{94}\\\color{purple}{9}\phantom{94}\end{array}$$$

Drag 9 down.

$$$\begin{array}{r}1\phantom{94}\\ 12\hspace{1pt})\overline{\hspace{1pt}21\color{red}{9}4}\\-\underline{12}\phantom{94}\\9\color{red}{9}\phantom{4}\end{array}$$$

Continue with 99.

How many 12s are there in 99? 99=12+12+12+12+12+12+12+12+3. There are eight 12s.

Multiply 12 and 8. Result is 96.

$$$\begin{array}{r}1\color{blue}{8}\phantom{4}\\ \color{blue}{12}\hspace{1pt})\overline{\hspace{1pt}2194}\\-\underline{12}\phantom{94}\\99\phantom{4}\\-\underline{\color{blue}{96}}\phantom{4}\end{array}$$$

Subtract 99 from 96. Result is 3.

$$$\begin{array}{r}18\phantom{4}\\12\hspace{1pt})\overline{\hspace{1pt}2194}\\-\underline{12}\phantom{94}\\\color{green}{99}\phantom{4}\\-\underline{\color{green}{96}}\phantom{4}\\\color{green}{3}\phantom{4}\end{array}$$$

Drag down 4.

$$$\begin{array}{r}18\phantom{4}\\ 12\hspace{1pt})\overline{\hspace{1pt}219\color{red}{4}}\\-\underline{12}\phantom{94}\\99\phantom{4}\\-\underline{96}\phantom{4}\\3\color{red}{4}\end{array}$$$

Finally, determine how many 12s in 34? 24=12+12+10. There are two 12s.

Multiply 12 by 2. Result is 24.

Subtract 34 from 24. Result is 10.

$$$\begin{array}{r}18\color{purple}{2}\\\color{purple}{12}\hspace{1pt})\overline{\hspace{1pt}2194}\\-\underline{12}\phantom{94}\\99\phantom{4}\\-\underline{96}\phantom{4}\\34\\-\underline{\color{purple}{24}}\\\color{cyan}{10}\end{array}$$$

We are done, because `10<12`.

So, remainder is 10, `2184=12*182+10` .

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

Exercise 1. Find remainder after division of 97 by 3.

Answer: 1 (`97=3*32+1`).

Exercise 2. Find remainder after division of 67 by 4.

Answer: 3 (`67=4*16+3`).

Exercise 3. Find remainder after division of 125 by 5.

Answer: 0 (`125=5*25`).

Exercise 4. Find remainder after division of 1653 by 24.

Answer: 21 (`1653=24*68+21`).

Exercise 5. Find remainder after division of 59540 by 447.

Answer: 446 (`59450=447*132+446`).