## Dividing Integers

### Learning Outcomes

• Divide integers with the same sign
• Divide integers with different signs
• Divide integers by -1

Division is the inverse operation of multiplication. So, $15\div 3=5$ because $5\cdot 3=15$ 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.

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

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:

1. $-27\div 3$
2. $-100\div \left(-4\right)$

Solution

 1. $-27\div 3$ Divide, noting that the signs are different and so the quotient is negative. $-9$
 2. $-100\div \left(-4\right)$ Divide, noting that the signs are the same and so the quotient is positive. $25$

### try it

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.

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

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

### Division by $-1$

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

$a\div \left(-1\right)=-a$

### example

Divide each of the following:

1. $16\div \left(-1\right)$
2. $-20\div \left(-1\right)$

### try it

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