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In this lesson, we will extend our knowledge of factors and multiples, **finding the greatest common factor** of two whole numbers less than or equal to 100, and the **least common multiple** of two whole numbers less than or equal to 12.

**See the fact file below for more information on the finding the GCF and LCM of two whole numbers or alternatively, you can download our 33-page Finding the GCF and LCM of Two Whole Numbers worksheet pack to utilise within the classroom or home environment.**

## Key Facts & Information

### FACTORS

- Factors are the numbers we multiply together to get another number or the product/multiple.
- A number can have several factors.
- Factors are whole numbers. They are never fractions.

### GETTING THE FACTORS OF A NUMBER

- We can start looking for factors by starting with 1 and the number. Let us use 120 as an example.
- We know that the product if we multiply 1 and a number is the number. Therefore, if we multiply 1 and 120, the answer is 120. Thus, 1 and 120 are factors of 120.
- 1, 120

- We cannot have a factor larger than the number itself.
- Now, starting from 1, we move forward.
- So, is 120 divisible by 2? Yes. And if we divide 120 by 2, the answer we will get is 60. Therefore, 2 and 60 are factors of 120.
- 1, 2, 60, 120

- Next, is 120 divisible by 3? Yes. If we divide 120 by 3, the answer we will get is 40. Thus, 3 and 40 are factors of 120.
- 1, 2, 3, 40, 60, 120

- Next, is 120 divisible by 4? Yes. And if we divide 120 by 4, the answer we will get is 30.
- 1, 2, 3, 4, 30, 40, 60, 120

- Next, we also know that 120 is divisible by 5. 5 times 24 is equal to 120.
- And, we also know that 120 is divisible by 6. 6 times 20 is equal to 120.
- Therefore, 5, 6, 20, and 24 are factors of 120.
- 1, 2, 3, 4, 5, 6, 20, 24, 30, 40, 60, 120

- Now, if we check if 120 is divisible by 7, the answer is no. Thus,

7 is not a factor of 120. - Moving on, 120 is divisible by 8. 120 divided by 8 is 15. But, we also know that 120 is not divisible by 9.
- Next, 120 is divisible by 10. And if we divide 120 by 10, the answer we will get is 12.
- 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

- As you can see, we still haven’t tried 11. But we know that 120 is not divisible by 11. Therefore, we can already end our list.
- Thus, the numbers above are the factors of 120.

### COMMON FACTORS

- Say we have worked out the factors of two whole numbers:
- Factors of 12: 1, 2, 3, 4, 6, and 12
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30

- The common factors of 12 and 30 are those found in both lists. Notice that 1, 2, 3, and 6 appear on the list of factors of 12 and 30. Therefore:

Common factors of 12 and 30: 1, 2, 3, and 6 - It is a common factor when it is a factor of two (or more) numbers.

### GREATEST COMMON FACTOR

- The Greatest Common Factor (GCF) is the largest of the common factors of two (or more) numbers.
- In our previous example, the common factors of 12 and 30 are 1, 2, 3, and 6. Therefore, the greatest common factor is 6.
- There are several different methods that can be used to find the GCF.
- Let’s take a look at some of the methods.
**METHOD 1: LIST OUT ALL THE FACTORS**- Example. Find the GCF of 64 and 96.
- Step 1. List the factors of each number.
- 64: 1, 2, 4, 8, 16, 32, 64
- 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
- Step 2. Look for factors that both lists have in common.
- 64: 1, 2, 4, 8, 16, 32, 64
- 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
- Step 3. Pick out the largest factor that both lists have in common.
- 64: 1, 2, 4, 8, 16, 32, 64
- 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
- Therefore, the GCF of 64 and 96 is 32.
**METHOD 2: UPSIDE DOWN DIVISION**- Example. Find the GCF of 280 and 144.
- Step 1. List the factors of each number.
- Step 2. Now, divide both numbers by a common factor. Since they are even, start with 2. The answer goes underneath the bar.
- Step 3. Continue to divide until you have two numbers that are relatively prime – they don’t have any common factors other than 1.
- 35 and 18 are relatively prime numbers.
- Step 4. Take all the factors on the side and multiply them together.
- This means that the GCF of 280 and 144 is 8.
**METHOD 3: PRIME FACTORIZATION**- We can also use prime factorization when finding the GCF of two numbers.
- Example. Find the GCF of 150 and 255.
- Step 1. Start by making factor trees for each of the numbers.
- Step 2. List out the prime factorization for each number.
- 150: 2 x 3 x 5 x 5
- 225: 3 x 3 x 5 x 5
- Step 3. Circle the prime factors that each number has in common.
- Step 4. Next, multiply the circled numbers together.
- 3 x 5 x 5 = 75
- This tells us that the GCF of 150 and 225 is 75.

### MULTIPLES

- Multiples, on the other hand, are different from factors.
- They are numbers we get after multiplying a number by an integer.
- WHAT ARE THE MULTIPLES OF 6?
- To find the multiples of 6, we need to multiply 6 by integers.
- Thus, based on the computations, the multiples of 6 are 6, 12, 18, 24, and 30.
- But that does not mean that these are the only multiples of 6.

66 is also a multiple of 6. - Therefore, numbers divisible by 6 are considered as multiples of 6.
- Note that we can also have negative results. How?
- 6 x -2 = -12

- This can happen since -2 is an integer.
- What are integers?
- Integers are numbers with no fractional part or decimals. Integers include counting numbers, zero, and the negative of counting numbers.

### COMMON MULTIPLES

- Say we have listed the first few multiples of 4 and 5, and the common multiples are those that are found in both lists.
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, …
- Notice that 20 and 40 appear in both lists. So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc.).

### LEAST COMMON MULTIPLE

- The Least Common Multiple (LCM) is the smallest of the common multiples.
- Example. Find the least common multiple of 4 and 10.
- 4: 4, 8, 12, 16, 20, …
- 10: 10, 20, …

- There’s a match at 20. Therefore, the LCM of 4 and 10 is 20.
- One way to easily find the LCM of two numbers (or more) is through prime factorization.
- Example. Find the LCM of 12 and 18.
- Step 1. List down their prime factors.
- 12: 2 x 2 x 3
- 18: 2 x 3 x 3

- Step 2. Write each number as a product of primes, matching primes vertically when possible.
- 12: 2 x 2 x 3
- 18: 2 x 3 x 3

- Step 3. Bring down the primes in each column. The LCM is the product of these factors.
- Notice that the prime factors of 12 and 18 are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that 35 is the LCM of 12 and 18.

**Finding the GCF and LCM of Two Whole Numbers Worksheets**

This is a fantastic bundle which includes everything you need to know about the finding the GCF and LCM of two whole numbers across 33 in-depth pages. These are** ready-to-use Finding the GCF and LCM of Two Whole Numbers worksheets that are perfect for teaching students about the factors and multiples, finding the greatest common factor of two whole numbers less than or equal to 100, and the least common multiple of two whole numbers less than or equal to 12. **

### Complete List Of Included Worksheets

- Lesson Plan
- Finding the GCF and LCM of Two Whole Numbers
- Finding Factors
- Finding Multiples
- Which is Which?
- Listing Method
- GCF Using Factor Trees
- Upside Down Division
- Find Through Listing
- LCM Using Factor Trees
- Just Some Questions
- Test Yourself!

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