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Factors and Multiples

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Factors of a Number A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. Factors of 24 1 X 24 2 X 12 3 X 8 4 X 6 Multiplication combinations to reach 24 {1, 2, 3, 4, 6, 8, 12, 24} A list of its factors

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**Greatest Common Factor (GCF)**

Greatest common factor is the greatest whole number that is a factor of two or more given whole numbers. Example next page

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**What is the greatest common factor of 36 and 48?**

Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36} 1 2 3 4 6 9 12 18 36 Factors of 48: {1, 2, 3, 4, 6, 8,12, 16, 24, 48}

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**What is the greatest common factor of 36 and 48?**

Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36} 1 2 3 4 6 12 Factors of 48: {1, 2, 3, 4, 6, 8,12, 16, 24, 48} 1 2 3 4 6 12 Which is the greatest common factor? Which factors are common to both numbers?

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**Greatest Common Factor (GCF)**

That was too much work. What happened if we forgot to include the 12 in one of the factor list? Want to learn a different way that’s easier? Duh!

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**Birthday Cake Method to find GCF**

Think of any number that divides evenly into both 36 and 48 (a factor of both) Divide both numbers by 2. For this example, I’ll use 2. 2

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**Birthday Cake Method to find GCF**

Think of any number that divides evenly into both 18 and 24 (a factor of both) Divide both numbers by 3. 6 8 For this example, I’ll use 3. 3 2

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**Birthday Cake Method to find GCF**

Continue this process until there is no other number besides 1 that is a factor of both numbers. 3 and 4 have no common factor except 1... so you can stop dividing. 3 4 2 6 8 3 2

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**2 x 3 x 2 = 12 Birthday Cake Method to find GCF 3 4 2 6 8 3 18 24 2**

The product of the numbers along the side of the cake is the GCF of 36 and 48. 2 x 3 x 2 = 12 3 4 2 6 8 3 2

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**8 x 2 = 16 Birthday Cake Method to find GCF**

Observe another example. Find the GCF of 64 and 80. 8 x 2 = 16 4 no common factor except 1 5 8 2

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Multiples of a Number The products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called the multiples of the number. 7 x 1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 For example: So, the multiples of 7 are 7, 14, 21, 28, and so on.

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**Multiples of a Number multiples of 2 are 2, 4, 6, 8, …**

Further examples: multiples of 2 are 2, 4, 6, 8, … multiples of 3 are 3, 6, 9, 12, … multiples of 4 are 4, 8, 12, 16, …

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Common Multiples Multiples that are common to two or more numbers are said to be common multiples. For example: Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, … Multiples of 3 are 3, 6, 9, 12, 15, 18, … So, common multiples of 2 and 3 are 6, 12, 18, … 6 12 18 12 18 6

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Example 2: Find the common multiples of 4 and 6. Solution: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, … Multiples of 6 are 6, 12, 18, 24, 30, 36, … So, the common multiples of 4 and 6 are 12, 24, 36, … 12 24 36 12 24 36

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**Least Common Multiple (LCM)**

The smallest common multiple of two or more numbers is called the least common multiple (LCM).

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**Least Common Multiple (LCM)**

To find the least common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM.

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**Least Common Multiple (LCM)**

Find the least common multiple of 6 and 9. Solution: List the multiples of 9 and stop when you find a multiple of 6. Multiples of 9 are 9, 18, … Multiples of 6 are 6, 12, 18, … LCM is 18

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**Birthday Cake Method to find LCM**

With the birthday cake method, you can find both the GCF and LCM at the same time! Let’s use a previous example…

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**Birthday Cake Method to find LCM**

Find the least common multiple of 36 and 48.

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**Birthday Cake Method to find LCM**

Find the product of the numbers going up and on top of the cake. Let’s make the birthday cake! 3 4 = 144 144 is the LCM of 36 and 48 x x 2 6 8 x 3 x 2

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**Birthday Cake Method to find LCM**

Another example: Find the LCM of 16 and 9. Do 16 and 9 have any common factors besides 1? No, they don’t. So their LCM is their product… ? 16 9 X = 144

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**Birthday Cake Method to find LCM**

Another example: Find the LCM of 5, 6 and 8. Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, … Stop at 120 as it is a multiple of both 5 and 6. Finding the answer by listing multiples of the largest number (8) would eventually lead you to the answer of 120.

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**Birthday Cake Method to find LCM**

The birthday cake method can be used to find the LCM of 5, 6 and 8. Watch! 5 3 4 x x = 120 x 2

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