Prime factorization is the process of breaking a whole number greater than 1 into a product of prime numbers. It matters because primes are the basic building blocks of multiplication, much like atoms are building blocks of matter. Once a number is written as prime factors, patterns in divisibility become easier to see.
This makes prime factorization useful for simplifying fractions, finding common factors, and comparing numbers.
Key Facts
- A prime number has exactly two positive factors: 1 and itself.
- A composite number has more than two positive factors and can be broken into smaller factors.
- 360 = 36 × 10 = 6 × 6 × 2 × 5 = 2 × 3 × 2 × 3 × 2 × 5.
- The prime factorization of 360 is 2^3 × 3^2 × 5.
- To find the GCF, multiply the common prime factors using the smaller exponents.
- To find the LCM, multiply all prime factors that appear using the larger exponents.
Vocabulary
- Prime number
- A prime number is a whole number greater than 1 with exactly two positive factors, 1 and itself.
- Composite number
- A composite number is a whole number greater than 1 that has more than two positive factors.
- Prime factorization
- Prime factorization is writing a number as a product of only prime numbers.
- Factor tree
- A factor tree is a diagram that repeatedly splits a number into factors until all final factors are prime.
- Exponent form
- Exponent form uses powers to write repeated factors more compactly, such as 2 × 2 × 2 = 2^3.
Common Mistakes to Avoid
- Stopping the factor tree too early is wrong because all final leaves must be prime numbers, not composite numbers like 4, 6, or 9.
- Forgetting repeated prime factors is wrong because every branch contributes to the final product, such as three 2s in 360 = 2^3 × 3^2 × 5.
- Writing factors in different orders as different answers is wrong because multiplication is commutative, so 2 × 3 × 2 and 2 × 2 × 3 represent the same prime factorization.
- Mixing up GCF and LCM is wrong because GCF uses shared primes with smaller exponents, while LCM uses all primes with larger exponents.
Practice Questions
- 1 Find the prime factorization of 84 and write your answer in exponent form.
- 2 Use prime factorization to find the GCF and LCM of 48 and 180.
- 3 Two students factor 72 as 8 × 9 and 6 × 12. Explain why both methods should lead to the same prime factorization.