Chapter Notes: Factors and Multiples
Have you ever shared a bag of candies equally with your friends? Or arranged toys into neat rows with the same number in each row? When you do this, you are working with factors and multiples. These are special numbers that help us understand how numbers fit together, divide evenly, and relate to each other. Learning about factors and multiples will help you solve all kinds of math problems, from sharing snacks to organizing groups in games!
What Are Factors?
A factor is a number that divides into another number exactly, with no remainder left over. When you multiply two whole numbers together, those two numbers are factors of the answer. Think of factors as the building blocks that multiply together to make a number.
For example, if you multiply 3 × 4, you get 12. This means that 3 and 4 are both factors of 12. They fit perfectly into 12 with no leftovers.
To find all the factors of a number, you look for all the whole numbers that divide into it evenly. Let's try finding the factors of 12:
- 1 × 12 = 12, so 1 and 12 are factors
- 2 × 6 = 12, so 2 and 6 are factors
- 3 × 4 = 12, so 3 and 4 are factors
The complete list of factors of 12 is: 1, 2, 3, 4, 6, and 12.
Example: Find all the factors of 18.
What are all the numbers that divide into 18 exactly?
Solution:
Start with 1: 1 × 18 = 18, so 1 and 18 are factors.
Try 2: 2 × 9 = 18, so 2 and 9 are factors.
Try 3: 3 × 6 = 18, so 3 and 6 are factors.
Try 4: 18 ÷ 4 = 4 remainder 2, so 4 is NOT a factor.
Try 5: 18 ÷ 5 = 3 remainder 3, so 5 is NOT a factor.
We already found 6, so we can stop here.
The factors of 18 are: 1, 2, 3, 6, 9, and 18.
The number 18 has six factors in total.
Factor Pairs
When you list factors, it helps to think in pairs. A factor pair is two numbers that multiply together to give you the original number. For 18, the factor pairs are:
- 1 and 18
- 2 and 9
- 3 and 6
Writing factor pairs helps you organize your work and makes sure you don't miss any factors.
Example: Find all the factor pairs of 24.
Which pairs of numbers multiply to make 24?
Solution:
1 × 24 = 24, so the pair is (1, 24).
2 × 12 = 24, so the pair is (2, 12).
3 × 8 = 24, so the pair is (3, 8).
4 × 6 = 24, so the pair is (4, 6).
5 does not divide 24 evenly.
The factor pairs of 24 are: (1, 24), (2, 12), (3, 8), and (4, 6).
This means the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Special Properties of Factors
Every whole number has at least two factors: 1 and itself. For example, 7 has factors 1 and 7. Some numbers have exactly two factors, and we call these prime numbers. Other numbers have more than two factors, and we call these composite numbers.
The number 1 is special. It has only one factor: itself. The number 1 is neither prime nor composite.
What Are Multiples?
A multiple of a number is the result you get when you multiply that number by any whole number. Think of multiples as the numbers you say when you skip count.
For example, the multiples of 3 are what you get when you count by threes:
- 3 × 1 = 3
- 3 × 2 = 6
- 3 × 3 = 9
- 3 × 4 = 12
- 3 × 5 = 15
So the first five multiples of 3 are: 3, 6, 9, 12, and 15. You can keep going forever because there are infinitely many multiples of any number.
Example: List the first six multiples of 4.
What are the first six numbers you get when you multiply 4 by 1, 2, 3, 4, 5, and 6?
Solution:
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
4 × 6 = 24
The first six multiples of 4 are: 4, 8, 12, 16, 20, and 24.
How Factors and Multiples Are Related
Factors and multiples are like opposite sides of the same coin. If 3 is a factor of 12, then 12 is a multiple of 3. They are two different ways to describe the same relationship between numbers.
Here's how to remember the difference:
- Factors are smaller numbers that divide into a bigger number.
- Multiples are bigger numbers that a smaller number divides into.
Example: Is 5 a factor of 30?
Is 30 a multiple of 5?How do 5 and 30 relate to each other?
Solution:
Check if 5 divides into 30 evenly: 30 ÷ 5 = 6 with no remainder.
Yes, 5 divides into 30 exactly, so 5 is a factor of 30.
Because 5 divides into 30, we also know that 30 is a multiple of 5.
Both statements are true and describe the same relationship.
Finding All Factors of a Number
When you need to find all the factors of a number, a good strategy is to test each whole number starting from 1. Check if the number divides evenly. If it does, write down both the divisor and the quotient because they are both factors.
Let's use a step-by-step process:
- Start with 1. Every number has 1 as a factor.
- Test 2, then 3, then 4, and so on.
- For each number that divides evenly, write down the factor pair.
- Stop when your factors start to repeat.
Example: Find all the factors of 36.
Which whole numbers divide into 36 with no remainder?
Solution:
1 × 36 = 36, so 1 and 36 are factors.
2 × 18 = 36, so 2 and 18 are factors.
3 × 12 = 36, so 3 and 12 are factors.
4 × 9 = 36, so 4 and 9 are factors.
5 does not divide 36 evenly.
6 × 6 = 36, so 6 is a factor.
We've reached the point where factors repeat, so we can stop.
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Common Factors
Sometimes you need to compare two numbers and find which factors they share. A common factor is a number that is a factor of both numbers you are looking at.
For example, let's find the common factors of 12 and 18:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
The numbers that appear in both lists are: 1, 2, 3, and 6. These are the common factors of 12 and 18.
Greatest Common Factor
The greatest common factor (GCF) is the largest number that is a factor of both numbers. From our example above, the common factors of 12 and 18 are 1, 2, 3, and 6. The greatest of these is 6, so the GCF of 12 and 18 is 6.
The GCF is very useful when simplifying fractions or dividing things into equal groups.
Example: Find the greatest common factor of 20 and 30.
What is the largest number that divides both 20 and 30 evenly?
Solution:
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Common factors: 1, 2, 5, 10
The greatest common factor is 10.
The number 10 is the largest number that divides both 20 and 30 with no remainder.
Common Multiples
Just like factors, you can compare multiples of two numbers. A common multiple is a number that is a multiple of both numbers.
For example, let's find some common multiples of 4 and 6:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60...
The numbers that appear in both lists include: 12, 24, 36, 48, and so on. These are common multiples of 4 and 6.
Least Common Multiple
The least common multiple (LCM) is the smallest number that is a multiple of both numbers. From the example above, the common multiples of 4 and 6 include 12, 24, 36, and 48. The smallest of these is 12, so the LCM of 4 and 6 is 12.
The LCM is very useful when adding or subtracting fractions with different denominators, or when solving problems about events that repeat at different intervals.
Example: Find the least common multiple of 5 and 8.
What is the smallest number that is a multiple of both 5 and 8?
Solution:
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56...
The first number that appears in both lists is 40.
The least common multiple is 40.
The number 40 is the smallest number that both 5 and 8 divide into evenly.
Using Factors and Multiples to Solve Problems
Understanding factors and multiples helps you solve many real-world problems. Here are some common situations where you might use these ideas:
- Sharing equally: If you have 24 cookies and want to share them equally among friends, the number of friends must be a factor of 24.
- Making arrays: If you arrange 18 chairs in equal rows, the number of rows and chairs per row are factor pairs of 18.
- Repeating patterns: If one alarm beeps every 4 minutes and another beeps every 6 minutes, they will beep together at times that are common multiples of 4 and 6.
Example: Maria has 28 stickers.
She wants to arrange them in equal rows with no stickers left over.What are the possible numbers of rows she can make?
Solution:
The number of rows must be a factor of 28.
Find all factors of 28: 1 × 28, 2 × 14, 4 × 7
The factors of 28 are: 1, 2, 4, 7, 14, 28
Maria can make 1, 2, 4, 7, 14, or 28 rows.
For example, she could make 4 rows with 7 stickers in each row.
Example: Two buses leave the station at the same time.
One bus returns every 12 minutes.
The other bus returns every 18 minutes.After how many minutes will both buses return to the station at the same time again?
Solution:
We need to find the least common multiple of 12 and 18.
Multiples of 12: 12, 24, 36, 48, 60...
Multiples of 18: 18, 36, 54, 72...
The first common multiple is 36.
Both buses will return together after 36 minutes.
Prime and Composite Numbers
When you study factors, you notice that some numbers have only two factors and others have more. This leads to an important way to classify numbers.
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Prime numbers cannot be divided evenly by any other numbers. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, and 23.
A composite number is a whole number greater than 1 that has more than two factors. Composite numbers can be divided evenly by at least one number besides 1 and themselves. Examples include: 4, 6, 8, 9, 10, 12, 14, 15, and 16.
The number 1 is special. It has only one factor (itself), so it is neither prime nor composite.
The number 2 is the only even prime number. All other even numbers are composite because they can be divided by 2.
Example: Is 17 a prime number or a composite number?
How many factors does 17 have?
Solution:
Check if any numbers besides 1 and 17 divide into 17 evenly.
17 ÷ 2 = 8 remainder 1 (not a factor)
17 ÷ 3 = 5 remainder 2 (not a factor)
17 ÷ 4 = 4 remainder 1 (not a factor)
Keep checking up to numbers close to 17 ÷ 2. None of them divide evenly.
The only factors of 17 are 1 and 17.
The number 17 is a prime number because it has exactly two factors.
Divisibility Rules
To find factors quickly, you can use divisibility rules. These are shortcuts that help you tell if one number divides into another without doing the full division.
| Divisor | Rule |
|---|---|
| 2 | The number ends in 0, 2, 4, 6, or 8 |
| 3 | The sum of the digits is divisible by 3 |
| 4 | The last two digits form a number divisible by 4 |
| 5 | The number ends in 0 or 5 |
| 6 | The number is divisible by both 2 and 3 |
| 9 | The sum of the digits is divisible by 9 |
| 10 | The number ends in 0 |
These rules save time and help you spot factors quickly.
Example: Is 72 divisible by 3?
Can you tell without dividing?
Solution:
Use the divisibility rule for 3: add the digits.
7 + 2 = 9
Since 9 is divisible by 3, the number 72 is also divisible by 3.
Yes, 72 is divisible by 3, and in fact 72 ÷ 3 = 24.
Summary of Key Ideas
Factors and multiples are two sides of the same relationship between numbers. Factors are numbers that divide into another number evenly. Multiples are the results you get when you multiply a number by whole numbers. Understanding how to find factors and multiples helps you solve problems about grouping, sharing, patterns, and repeating events.
The greatest common factor (GCF) is the largest number that divides two or more numbers. The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. Both the GCF and LCM are very useful in working with fractions and solving real-world problems.
Prime numbers have exactly two factors, while composite numbers have more than two. Recognizing prime and composite numbers helps you understand the building blocks of all numbers. Using divisibility rules makes finding factors faster and easier.
