### Learning Outcomes

- Divide integers

Division is the inverse operation of multiplication. So, [latex]15\div 3=5[/latex] because [latex]5\cdot 3=15[/latex] In words, this expression says that [latex]\mathbf{\text{15}}[/latex] can be divided into [latex]\mathbf{\text{3}}[/latex] groups of [latex]\mathbf{\text{5}}[/latex] each because adding five three times gives [latex]\mathbf{\text{15}}[/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

[latex]\begin{array}{ccccc}5\cdot 3=15\text{ so }15\div 3=5\hfill & & & & -5\left(3\right)=-15\text{ so }-15\div 3=-5\hfill \\ \left(-5\right)\left(-3\right)=15\text{ so }15\div \left(-3\right)=-5\hfill & & & & 5\left(-3\right)=-15\text{ so }-15\div -3=5\hfill \end{array}[/latex]

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

### Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same signs | Quotient |
---|---|

•Two positives
•Two negatives |
Positive
Positive |

Different signs | Quotient |
---|---|

•Positive & negative
•Negative & positive |
Negative
Negative |

Remember, you can always check the answer to a division problem by multiplying.

### example

Divide each of the following:

- [latex]-27\div 3[/latex]
- [latex]-100\div \left(-4\right)[/latex]

Solution

1. | |

[latex]-27\div 3[/latex] | |

Divide, noting that the signs are different and so the quotient is negative. | [latex]-9[/latex] |

2. | |

[latex]-100\div \left(-4\right)[/latex] | |

Divide, noting that the signs are the same and so the quotient is positive. | [latex]25[/latex] |

### try it

Just as we saw with multiplication, when we divide a number by [latex]1[/latex], the result is the same number. What happens when we divide a number by [latex]-1?[/latex] Let’s divide a positive number and then a negative number by [latex]-1[/latex] to see what we get.

[latex]\begin{array}{cccc}8\div \left(-1\right)\hfill & & & -9\div \left(-1\right)\hfill \\ -8\hfill & & & 9\hfill \\ \hfill \text{-8 is the opposite of 8}\hfill & & & \hfill \text{9 is the opposite of -9}\hfill \end{array}[/latex]

When we divide a number by [latex]-1[/latex] we get its opposite.

### Division by [latex]-1[/latex]

Dividing a number by [latex]-1[/latex] gives its opposite.

[latex]a\div \left(-1\right)=-a[/latex]

### example

Divide each of the following:

- [latex]16\div \left(-1\right)[/latex]
- [latex]-20\div \left(-1\right)[/latex]

### try it

Watch the following video for more examples of how to divide integers with the same and different signs.