Transcript Factorization - Olathe School District

Factorization
Greatest Common Factor
Least Common Multiple
Why Factor?
Factors allow you to break composite
numbers down to their component parts.
 Factors are used to simplify fractions.
 Factors are used to identify the greatest
common factor (GCF) and the least
common multiple (LCM).
 A number can be written as the product of
its prime factors.

Find Factor Pairs for 24
Start with 1: 1 x 24 = 24
 2: 2 x 12 = 24
 3: 3 x 8 = 24
 4: 4 x 6 = 24
 6: 6 x 4 = 24
 8: 8 x 3 = 24
 12: 12 x 2 = 24
 24: 24 x 1 = 24

The commutative
property shows the
same pairs:
1 x 24 = 24 x 1
2 x 12 = 12 x 2
3x8=8x3
4x6=6x4
Prime Factorization
Prime numbers are numbers that have
only one and the number as factors.
 “1” is neither prime or composite.
 Composite numbers can be written as
products of their prime factors.

Factorization
Upside Down Division
Prime factorization of 24
2 24
2 12
2 6
3
Prime Factorization is 2 x 2 x 2 x 3
Prime factorization of 28
2 28
2 14
7
Prime Factorization is 2 x 2 x 7
Prime factorization of 15
3 15
5
Prime Factorization is 3 x 5
Prime factor 40
2 40
2 20
2 10
5
Prime Factorization is 2 x 2 x 2 x 5
Greatest Common Factor (GCF)

Definition:
– The largest factor that divides evenly into 2 or
more numbers.

Examples:
– 3 and 7 have no common factors other than 1
– 4 and 6 have the greatest common factor of 2
– 6 and 24 have a greatest common factor of 6
– 10 and 15 have a GCF of 5
Greatest Common Factor
Use Factorization
Prime factor 28 and 16
to find the GCF
2 28
2 16
2 8
2 14
2
4
7
2
Prime Factorization
of 28: 2 x 2 x 7
Prime Factorization
of 16: 2 x 2 x 2 x 2
Prime factor 28 and 16
2 28
2 14
7
Prime Factorization
of 28: 2 x 2 x 7
2 16
2 8
2 4
2
Prime Factorization
of 16: 2 x 2 x 2 x 2
Factor both numbers 28 and 16
at the same time
2 28
2 14
7
16
8
4
The GCF is the product of the
common prime factors: 2 x 2 = 4
Factor both numbers: 24 and 30
2 24
3 12
4
30
15
5
The GCF is the product of the
common prime factors: 2 x 3 = 6
Factor both numbers: 28 and 42
G
2 28
7 14
2
42
21
3
The GCF is the product of the
common prime factors: 2 x 7 = 14
Factor both numbers: 12 and 30
G
2 12
3 6
2
30
15
5
The GCF is the product of the
common prime factors: 2 x 3 = 6
Least Common Multiple
Use Factorization
Factor both numbers 28 and 16
at the same time
2 28
2 14
7
16
8
4
The LCM is the product of ALL the
factors: 2 x 2 x 7 x 4 = 112
Factor both numbers: 24 and 30
2 24
3 12
4
30
15
5
The LCM is the product of ALL of
the factors: 2 x 3 x 4 x 5 = 120
Factor both numbers: 28 and 42
2 28
7 14
2
42
21
3
The LCM is the product of ALL of
the factors: 2 x 7 x 2 x 3 = 84
Factor both numbers: 12 and 30
2 12
3 6
2
30
15
5
The LCM is the product of ALL of
the factors: 2 x 3 x 2 x 5 = 60
Greatest Common Factor
and
Least Common Multiple
Factor both numbers: 10 and 30
G
5 10
2 2
1
30
6
3
GCF: 5 x 2 = 10
LCM: 5 x 2 x 1 x 3 = 30
Factor both numbers: 12 and 40
G
2 12
2 6
3
40
20
10
GCF: 2 x 2 = 4
LCM: 2 x 2 x 3 x 10 = 120
Factor both numbers: 12 and 40
G
2 12
2 6
3
40
20
10
GCF: 2 x 2 = 4
LCM: 2 x 2 x 3 x 10 = 120
Simplifying Fractions
Greatest Common Factor
Simplify 24/30
2 24
3 12
4
30
15
5
The fraction 24/30
simplifies to 4/5
(The GCF was used: 24/6 = 4, 30/6 = 5)
Simplify 28/42
2 28
7 14
2
42
21
3
The fraction 28/42
simplifies to 2/3
(The GCF was used: 28/14 = 2, 42/14 = 3)
Simplify 12/40
2 12
2 6
3
40
20
10
The fraction 12/40
simplifies to 3/10
(The GCF was used: 12/4 = 3, 40/4 = 10)