Factoring helps us understand how numbers are built. Instead of only thinking about a number as a single amount, we can break it apart into smaller whole-number pieces that multiply together. In this topic, we explore factors, factorizations, prime numbers, and composite numbers as tools for understanding number structure.

A factor is a whole number that divides evenly into another whole number. For example, 3 is a factor of 12 because 3×4=12. A factorization is one way to write a number as a product of factors, such as 12=3×4 or 12=2×2×3. Numbers that have exactly two factors, 1 and themselves, are called prime numbers, while numbers with more than two factors are called composite numbers.

We will also look at methods for finding and organizing factors. The Sieve of Eratosthenes helps identify prime numbers by eliminating multiples. Factor trees help break composite numbers into prime factors. The Fundamental Theorem of Arithmetic tells us that every whole number greater than 1 can be written as a product of prime numbers in exactly one way, except for the order of the factors.

Finally, we will use prime factorization to find all factors of a number. This helps show that factors are not random; they come from the different combinations of a number’s prime factors. As future teachers, understanding factorization deeply can help you support students with multiplication, division, fractions, least common multiples, greatest common factors, and later algebra.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define factor, factorization, prime number, and composite number.

  • Identify whether a number is prime or composite.

  • Use the Sieve of Eratosthenes to find prime numbers.

  • Create factor trees to find prime factorizations.

  • Explain the Fundamental Theorem of Arithmetic in student-friendly language.

  • Use prime factorization to find all factors of a number.

  • Connect factorization to multiplication, division, and number structure.

Key Vocabulary

  • Factor - A whole number that divides evenly into another number.

  • Factorization - Writing a number as a product of factors.

  • Prime Number - A whole number greater than 1 with exactly two factors: 1 and itself.

  • Composite Number - A whole number greater than 1 with more than two factors.

  • Multiple - The result of multiplying a number by a whole number.

  • Sieve of Eratosthenes - A method for finding prime numbers by crossing out multiples.

  • Factor Tree - A diagram that breaks a number into smaller factors until only primes remain.

  • Prime Factorization - Writing a number as a product of prime numbers.

  • Fundamental Theorem of Arithmetic - The idea that every whole number greater than 1 has one unique prime factorization, except for order.

Example Strategies

  • Find factors - Factors of 18 - 1, 2, 3, 6, 9, 18

  • Decide prime or composite - 17 - Prime, because its only factors are 1 and 17.

  • Decide prime or composite - 21 - Composite, because 3 × 7 =21.

  • Use a factor tree - 36 = 4 × 9 = 2 × 2 × 3 × 3.

  • Write prime factorization - 60 = 2 × 2 × 3 × 5.

  • Find all factors from prime factorization - 12 = 2 × 2 × 3 - Use combinations of 1, 2, 4 with 1, 3: 1, 2, 3, 4, 6, 12 .

Teacher Connection

Factoring is an important foundation for many later math topics. Students use factors when simplifying fractions, finding common denominators, identifying greatest common factors, finding least common multiples, and eventually factoring algebraic expressions.

As a future teacher, it is helpful to emphasize that factoring is about structure. A number like 36 is not just “thirty-six.” It can also be seen as 4 × 9, 6 × 6, 3 × 12, or 2 × 2 × 3 × 3. Each representation can reveal something useful.

Helpful teacher questions include:

  • What numbers multiply to make this number?

  • How do you know this number is prime or composite?

  • Can you find the factors in pairs?

  • Are the ends of your factor tree all prime?

  • Would a different factor tree give the same prime factorization?

  • How can the prime factorization help us find every factor?

Quick Reflection Question

Why does the Fundamental Theorem of Arithmetic matter? How does knowing that every number has a unique prime factorization help us understand factors more clearly?