Mathematics » Introducing Integers » Multiply and Divide Integers

# Dividing Integers

## Dividing Integers

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

$$\begin{array}{ccccc}5·3=15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷3=5\hfill & & & & -5\left(3\right)=-15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷3=-5\hfill \\ \left(-5\right)\left(-3\right)=15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15÷\left(-3\right)=-5\hfill & & & & 5\left(-3\right)=-15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15÷-3=5\hfill \end{array}$$

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 signsQuotient
•Two positives

•Two negatives

Positive

Positive

Different signsQuotient
•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:

1. $$−27÷3$$
2. $$\phantom{\rule{0.2em}{0ex}}-100÷\left(-4\right)$$

### Solution

 $$–27÷3$$ Divide, noting that the signs are different and so the quotient is negative. $$–9$$
 $$–100÷\left(–4\right)$$ Divide, noting that the signs are the same and so the quotient is positive. $$25$$

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

$$\begin{array}{cccc}8÷\left(-1\right)\hfill & & & -9÷\left(-1\right)\hfill \\ -8\hfill & & & 9\hfill \\ \hfill \text{−8 is the opposite of 8}\hfill & & & \hfill \text{9 is the opposite of −9}\hfill \end{array}$$

When we divide a number by, $$-1$$ we get its opposite.

### Division by -1

Dividing a number by $$-1$$ gives its opposite.

$$a÷\left(-1\right)=\mathit{\text{−a}}$$

## Example

Divide each of the following:

1. $$\phantom{\rule{0.2em}{0ex}}16÷\left(-1\right)\phantom{\rule{1em}{0ex}}$$
2. $$\phantom{\rule{0.2em}{0ex}}-20÷\left(-1\right)$$

### Solution

 $$16÷\left(–1\right)$$ The dividend, 16, is being divided by –1. $$–16$$ Dividing a number by –1 gives its opposite. Notice that the signs were different, so the result was negative.
 $$–20÷\left(–1\right)$$ The dividend, –20, is being divided by –1. $$20$$ Dividing a number by –1 gives its opposite.

Notice that the signs were the same, so the quotient was positive.