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Greatest Common Factor (GCF) and Least Common Multiple (LCM). Prime Factorization. Prime Factorization is when a number is expressed as a product of prime factors. You can use a factor tree to find the prime factorization of a number. Example: Find the prime factorization of 12. 12. 2. 6.

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prime factorization
Prime Factorization

Prime Factorization is when a number is expressed as a product of prime factors.

You can use a factor tree to find the prime factorization of a number.

Example: Find the prime factorization of12.

12

2

6

2

3

2

factoring monomials
Factoring Monomials

You can use prime factorization to factor monomials. A monomial is in factored form when it is expressed as the product of prime numbers and variables and no variable has an exponent greater than 1.

Example: Factor 100mn3.

100

mn3

= mnnn

10

10

5

2

5

2

finding a gcf through reasoning
Finding a GCF through Reasoning

Find the greatest common factor of:

8, 28, 12

Factors of 8:

Factors of 28:

Factors of 12:

Find the largest number that divides all three numbers.

The GCF is 4.

This process could be long and tedious if the numbers are large. A more efficient method is desirable.

greatest common factor gcf
Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the greatest expression that is a factor of the original expressions.

Procedure to find the GCF of two or more terms:

  • Factor each monomial.
  • The GCF is the product of the common factors.
examples
Examples

Find the GCF of each set of monomials:

  • 24, 60, and 72
  • 15 and 8
  • 15a2b, 9ab2, and 18ab

Step One: Factor each Monomial.

Step Two: Find the common factors.

There is nothing in common, so the GCF is…

Step Three: Multiply the factors in common.

finding a lcm through reasoning
Finding a LCM through Reasoning

Find the least common multiple of:

8, 28, 12

Multiples of 8:

Multiples of 28:

Multiples of 12:

Find the smallest number that is a multiple of all three.

The LCM is 168.

This process could be long and tedious if the numbers are large. A more efficient method is desirable.

least common multiple lcm
Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the least number that is a common multiple of two or more expressions.

Procedure to find the LCM of two or more terms:

  • Factor each monomial.
  • Find the greatest number of times each factor appears in each factorization.
  • The LCM is the product of (2).
examples1
Examples

Find the GCF of each set of monomials:

  • 18, 30, and 105
  • 15a2b and 27b3

Step One: Factor each Monomial.

Step Two: Find the greatest number of times each factor appears in each factorization.

Step Three: Multiply the result of (2).