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Common multiples

Multiples of 6

Multiples of 8

12

60

6

18

54

66

102…

24

24

48

48

96

96

72

8

16

32

40

56

64

72

80

88

104 …

30

42

78

90

84

36

The least common multiple

The least common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers.

We can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.

For example:

100,

Multiples of 20 are :

20,

40,

60,

80,

120, . . .

25,

50,

75,

100,

125, . . .

Multiples of 25 are :

The LCM of 20 and 25 is

100.

The least common multiple

What is the least common multiple (LCM) of 8 and 10?

The first ten multiples of 8 are:

8,

16,

24,

32,

40,

48,

56,

64,

72,

80.

The first ten multiples of 10 are:

10,

20,

30,

40,

50,

60,

70,

80,

90,

100.

The least common multiple (LCM) of 8 and 10 is

40.

The least common multiple

5

5

and .

Add together

4

4

12

12

9

9

× 4

× 3

31

+

=

+

36

36

36

× 4

× 3

We use the least common multiplewhen adding and subtracting fractions.

The LCM of 9 and 12 is 36.

16

15

=

The greatest common factor

The greatest common factor (or GCF) of two numbers is the largest number that is a factor of both numbers.

We can find the greatest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.

For example:

Factors of 36 are :

1,

2,

3,

4,

6,

9,

12,

18,

36.

Factors of 45 are :

1,

3,

5,

9,

15,

45.

The GCF of 36 and 45 is

9.

The greatest common factor

What is the greatest common factor (GCF) of 24 and 30?

The factors of 24 are:

1,

2,

3,

4,

6,

8,

12,

24.

The factors of 30 are:

1,

2,

3,

5,

6,

10,

15,

30.

The greatest common factor (GCF) of 24 and 30 is

6.

The greatest common factor

36

36

48

48

÷12

÷12

We use the greatest common factor when simplifying fractions.

Simplify the fraction .

The GCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12.

3

=

4

Using prime factors to find the GCF and LCM

2

60

2

294

1

1

We can use the prime factorization to find the GCF and LCM of larger numbers.

Find the GCF and the LCM of 60 and 294.

2

30

3

147

3

15

7

49

5

5

7

7

60 = 2 × 2 × 3 × 5

294 = 2 × 3 × 7 × 7

Using prime factors to find the GCF and LCM

60 = 2 × 2 × 3 × 5

294 = 2 × 3 × 7 × 7

60

294

2

7

2

3

5

7

GCF of 60 and 294 =

2 × 3

= 6

LCM of 60 and 294 =

2 × 5 × 2 × 3 × 7 × 7 =

2940

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