Factors and Multiples. Factors of a Number A number may be made by multiplying two or more other...

Factors and MultiplesFactors and Multiples

Factors of a NumberFactors of a NumberA number may be made A number may be made

by multiplying two or by multiplying two or more other numbers more other numbers together. The numbers together. The numbers that are multiplied that are multiplied together are called together are called factors of the final factors of the final number. number.

Factors of 24

1 X 242 X 123 X 84 X 6

{1, 2, 3, 4, 6, 8, 12, 24}A list of its factors

Multiplicationcombinations to

reach 24

Greatest Common Factor Greatest Common Factor (GCF)(GCF)

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

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

Factors of 36: Factors of 36:


{1, 2, 3, 4, 6, 9, 12, 18, 36}361 2 183 124 96

{1, 2, 3, 4, 6, 8,12, 16, 24, 48}



{1, 2, 3, 4, 6, 9, 12, 18, 36}

{1, 2, 3, 4, 6, 8,12, 16, 24, 48}

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

Which factors are common to both numbers?

12

12

6

6

4

4

3

3

2

2

1

1

Which is the greatest common factor?

That was too much work. What That was too much work. What happened if we forgot to happened if we forgot to include the 12 in one of the include the 12 in one of the factor list?factor list?

Greatest Common Factor Greatest Common Factor (GCF)(GCF)

Want to learn a different way that’s easier?

Birthday Cake Method to find Birthday Cake Method to find GCFGCF

36 48

Think of any number that divides evenly into both 36 and 48

(a factor of both)

For this example, I’ll use 2.

2

Divide both numbers by 2.

18 24

36 48218 24


Think of any number that divides evenly into both 18 and 24

(a factor of both)

For this example, I’ll use 3.

3

Divide both numbers by 3.

6 8

36 48218 24


3

6 8

Continue this process until there is no other number besides 1 that is a factor

of both numbers.

23 4

3 and 4 have no common factor except 1... so you can stop dividing.

36 48218 243

6 823 4


The product of the numbers along the side of the cake is the GCF

of 36 and 48.

The product of the numbers along the side of the cake is the GCF

of 36 and 48.2 x 3 x 2 = 122 x 3 x 2 = 12


Observe another example. Find theGCF of 64 and 80.

64 80232 408

4 58 x 2 = 16

no commonfactor except 1

Multiples of a NumberMultiples of a NumberThe products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called the multiples of the number.

For example: 7 x 1 = 77 x 2 = 147 x 3 = 217 x 4 = 28

So, the multiples of 7 are 7, 14, 21, 28, and so on.

Further examples:

multiples of 2 are 2, 4, 6, 8, …



Multiples of a NumberMultiples of a Number

Common MultiplesCommon MultiplesMultiples that are common to two or more Multiples that are common to two or more

numbers are said to be numbers are said to be common multiplescommon multiples. . For example:For example:

Multiples of 2 are 2, 4, Multiples of 2 are 2, 4, 66, 8, 10, , 8, 10, 1212, 14, 16, , 14, 16, 1818, , ……

Multiples of 3 are 3, Multiples of 3 are 3, 66, 9, , 9, 1212, 15, , 15, 1818, …, …

So, common multiples of 2 and 3 are 6, 12, 18, So, common multiples of 2 and 3 are 6, 12, 18, ……

66 1212

12121818

181866

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, …

363636362424

2424

1212

1212

Least Common Multiple Least Common Multiple (LCM)(LCM)

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

Least Common Multiple (LCM)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.

Least Common Multiple (LCM)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

Birthday Cake Method to find Birthday Cake Method to find LCMLCM

With the birthday cake method, you can find both the GCF and LCM at the same time!

Let’s use a previous example…


Find the least common multiple of 36 and 48.


36 48218 243

6 823 4

Let’s make the birthday cake!Find the product of thenumbers going up and on top of the cake.

x

x

x x = 144144 is the

LCMof 36 and

48


Another example: Find the LCM of 16 and 9.

16 9

Do 16 and 9 have any common factors besides 1?

?

No, they don’t. So their LCM is their product…

X = 144


Another example: Find the LCM of 5, 6 and 8.

Finding the answer by listing multiples of the largest number (8)would eventually lead you to the answer of 120.

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.


The birthday cake method can be used to find the LCM of 5, 6 and 8. Watch!

5 6 823 45x x x = 120

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