Lesson Objectives
• Demonstrate an understanding of integers
• Learn the sign rules for multiplying & dividing integers
• Learn how to find the product of two or more integers
• Learn how to find the quotient of two integers

## How to Multiply & Divide Integers

### How to Multiply two Integers

Up to this point, we have only performed the multiplication operation with whole numbers. From our previous studies, we should know that the product of two positive numbers is positive:
(+3) x (+5) = +15
(+8) x (+9) = +72
What happens when we start multiplying with negative numbers? When multiplying with two integers only, the rules are very simple:
The product is positive if the two signs are the same:
(+) x (+) = (+)
(-) x (-) = (+)
The product is negative if the two signs are different:
(+) x (-) = (-)
(-) x (+) = (-)
To multiply two integers together, we multiply absolute values and then attach the correct sign based on the sign rules. Let's take a look at a few examples.
Example 1: Find each product.
(-5) x (-8)
• Multiply the absolute values
• 5 x 8 = 40
• Attach the correct sign based on the sign rules:
• Both factors (-5) and (-8) are negative
• (-) x (-) = (+)
• +40
(-5) x (-8) = 40
Example 2: Find each product.
(-12) x (13)
• Multiply the absolute values
• 12 x 13 = 156
• Attach the correct sign based on the sign rules:
• One factor is negative (-12) and the other is positive (13)
• (-) x (+) = (-)
• -156
(-12) x (13) = -156
Example 3: Find each product.
(-21) x (18)
• Multiply the absolute values
• 21 x 18 = 378
• Attach the correct sign based on the sign rules:
• One factor is negative (-21) and the other is positive (18)
• (-) x (+) = (-)
• -378
(-21) x (18) = -378

### Multiplying more than two Integers

What happens when we want to multiply more than two integers? There are two ways to complete this task:
1. We can multiply pairs of numbers until we have our final product. Remember, multiplication is commutative, so we can rearrange the multiplication into whatever order is easiest.
2. We can determine the sign of the product first and then multiply absolute values. This is generally a little faster since you only deal with the sign once:
• Each pair of negative signs yields a positive. For every 2 negatives, you will get a positive result. This means if there is an even number (2,4,6,8,10,…) of negatives, you will get a positive product.
• When there is an odd number (1,3,5,7,9,…) of negatives, you will get a negative product.
Let's look at a few examples.
Example 4: Find each product.
(-5) x (4) x (-2) x (-8) x (3)
• If we use the second method, we can count the number of negatives involved. Here we have 3 negative numbers: (-5), (-2), and (-8). Since 3 is an odd number, the result of the multiplication will be negative.
• Now that we know the sign, we can multiply absolute values in any order that we would like.
• 5 x 4 x 2 x 8 x 3 = 20 x 2 x 8 x 3 = 40 x 8 x 3 = 320 x 3 = 960
• Attach the (-) sign to the answer above
• -960
(-5) x (4) x (-2) x (-8) x (3) = -960
Example 5: Find each product.
(-1) x (-6) x (-5) x (4) x (-2)
• If we use the second method, we can count the number of negatives involved. Here we have 4 negative numbers: (-1), (-6), (-5), and (-2). Since 4 is an even number, the result of the multiplication will be positive.
• Now that we know the sign, we can multiply absolute values in any order that we would like.
• 1 x 6 x 5 x 4 x 2 = 6 x 5 x 4 x 2 = 30 x 4 x 2 = 120 x 2 = 240
• The answer will be positive (+)
• +240
(-1) x (-6) x (-5) x (4) x (-2) = 240

### How to Divide Integers

Once you understand how to multiply two integers, division will be easy. We divide integers using the same rules as for multiplication.
If the signs are the same, the quotient will be positive:
(+) ÷ (+) = (+)
(-) ÷ (-) = (+)
If the signs are different, the quotient will be negative:
(+) ÷ (-) = (-)
(-) ÷ (+) = (-)
To divide integers, we divide absolute values and then attach the correct sign based on the sign rules. Let's look at a few examples:
Example 6: Find each quotient.
(-15) ÷ (-3)
• Divide the absolute values
• 15 ÷ 3 = 5
• Attach the correct sign based on the sign rules:
• Both factors (-15) and (-3) are negative
• (-) ÷ (-) = (+)
• +5
(-15) ÷ (-3) = 5
Example 7: Find each quotient.
(-28) ÷ (7)
• Divide the absolute values
• 28 ÷ 7 = 4
• Attach the correct sign based on the sign rules:
• One factor is negative (-28) and the other is positive (7)
• (-) ÷ (+) = (-)
• -4
(-28) ÷ (7) = -4

#### Skills Check:

Example #1

Find each product.

(-26) x (-47)

A
-1,222
B
1,222
C
877
D
-877
E
1524

Example #2

Find each product.

14 x (-11) x 7

A
-1078
B
1078
C
1848
D
-1844
E
1355

Example #3

Find each quotient.

1,617 ÷ (-33)

A
-37
B
37
C
-65
D
-49
E
49         