Factors
Prime factorization expresses any whole number as a unique product of prime numbers, a consequence of the Fundamental Theorem of Arithmetic. In Grade 6 Saxon Math Course 1, students use factor trees to systematically break numbers down: 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5. This skill is directly applied to finding GCF, finding LCM, and reducing fractions. The uniqueness of prime factorization means every number has exactly one such representation.
Key Concepts
Contextual Explanation Think of prime numbers as the ultimate atoms of the number world. Every whole number that isn't prime (a composite number) can be built by multiplying a unique set of primes. This is called prime factorization and is a super powerful tool for simplifying complex math problems down the road. Full Example Problem : Show which prime numbers we multiply to make the number 36. Solution : We break 36 down into only prime factors. Start with the smallest prime, 2: $36 = 2 \times 18$. Break down 18: $18 = 2 \times 9$. Break down 9. It's not divisible by 2, so try the next prime, 3: $9 = 3 \times 3$. All factors are now prime. The prime factorization of 36 is $2 \times 2 \times 3 \times 3$ .
Common Questions
What is prime factorization?
Writing a number as a product of prime numbers only. Example: 36 = 2 × 2 × 3 × 3 = 2² × 3².
How do you use a factor tree for 120?
120 → 2 × 60 → 2 × 2 × 30 → 2 × 2 × 2 × 15 → 2 × 2 × 2 × 3 × 5. Result: 2³ × 3 × 5.
Why is prime factorization useful for simplifying fractions?
It reveals shared prime factors in numerator and denominator so you can cancel them to reduce the fraction in one step.
Is 1 a prime number?
No. A prime number has exactly two factors (1 and itself). The number 1 has only one factor, so it is neither prime nor composite.
What guarantees a number has a unique prime factorization?
The Fundamental Theorem of Arithmetic states every integer greater than 1 has exactly one prime factorization (order of factors aside).