Prime Factors and Greatest Common Factors

Prime Factors

• A prime factor is a prime number that divides a given number exactly without leaving a remainder.
• Every positive integer can be expressed as a product of prime factors in a unique way, known as the prime factorization.
• Prime factors are used to simplify fractions, find greatest common factors, and calculate least common multiples.

Greatest Common Factor (GCF)

• The greatest common factor (GCF) is the largest number that divides two or more numbers exactly without leaving a remainder.
• To find the GCF, list the prime factors of each number and identify the common prime factors.
• The GCF is the product of the common prime factors, each raised to the lowest power.

Multiples Patterns

• Multiples of a number are the products of that number and the counting numbers (1, 2, 3, ...).
• Multiples patterns can be used to identify relationships between numbers and to solve problems involving multiples.
• Examples of multiples patterns include:
  • The multiples of 3 are 3, 6, 9, 12, ...
  • The multiples of 4 are 4, 8, 12, 16, ...

Least Common Multiple (LCM)

• The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.
• To find the LCM, list the multiples of each number and identify the first multiple that appears in both lists.
• The LCM can also be found using prime factorization: list the prime factors of each number, and multiply the highest power of each prime factor that appears in both lists.

Factor Trees

• A factor tree is a diagram that shows the prime factors of a number.
• To create a factor tree, start with the number and divide it by the smallest prime factor.
• Continue dividing each factor by the smallest prime factor until only prime factors remain.
• The prime factors can be multiplied together to find the original number.

Prime Factors

• A prime factor is a prime number that divides a given number exactly without leaving a remainder.
• Every positive integer can be expressed as a product of prime factors in a unique way, known as the prime factorization.
• Prime factors are used to simplify fractions, find greatest common factors, and calculate least common multiples.

Greatest Common Factor (GCF)

• The GCF is the largest number that divides two or more numbers exactly without leaving a remainder.
• To find the GCF, list the prime factors of each number and identify the common prime factors.
• The GCF is the product of the common prime factors, each raised to the lowest power.

Multiples Patterns

• Multiples of a number are the products of that number and the counting numbers (1, 2, 3,...).
• Multiples patterns can be used to identify relationships between numbers and to solve problems involving multiples.
• Examples of multiples patterns include:
  • The multiples of 3 are 3, 6, 9, 12,...
  • The multiples of 4 are 4, 8, 12, 16,...

Least Common Multiple (LCM)

• The LCM is the smallest number that is a multiple of two or more numbers.
• To find the LCM, list the multiples of each number and identify the first multiple that appears in both lists.
• The LCM can also be found using prime factorization: list the prime factors of each number, and multiply the highest power of each prime factor that appears in both lists.

Factor Trees

• A factor tree is a diagram that shows the prime factors of a number.
• To create a factor tree, start with the number and divide it by the smallest prime factor.
• Continue dividing each factor by the smallest prime factor until only prime factors remain.
• The prime factors can be multiplied together to find the original number.