# Fundamental Mathematics/Arithmetic/Division

## Operation

${\displaystyle {\frac {A}{B}}=}$ A ÷ B

Where

${\displaystyle A}$
${\displaystyle B}$
${\displaystyle C}$

## Divisibility

Division without remainder

${\displaystyle {\frac {A}{B}}=C}$ without remainder then ${\displaystyle A=BC}$

Division with remainder

${\displaystyle {\frac {A}{B}}=C}$ with remainder R then ${\displaystyle A=BC+R}$

## Division Table

 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144

## Example

• Definition: ${\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}$ , whenever ${\displaystyle b\neq 0}$ .

Let's look at an example to see how these rules are used in practice. Of course, the above is much longer than simply cancelling ${\displaystyle x+3}$ out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:

${\displaystyle {\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{2}}=1\times {\frac {x+2}{2}}={\frac {x+2}{2}}}$ .

The correct simplification is

${\displaystyle {\frac {2\times (x+2)}{2}}=\left(2\times {\frac {1}{2}}\right)\times (x+2)=1\times (x+2)=x+2}$ ,

where the number ${\displaystyle 2}$ cancels out in both the numerator and the denominator.