Lecture Notes: Prime Factors and Factor Trees

Introduction to Prime Factors

• Prime Factor: A factor of a number that is also a prime number.
  • Example: For the number 12, the factors are 1, 2, 3, 4, 6, and 12.
  • Prime factors of 12 are 2 and 3 because both are prime numbers.

Writing Numbers as a Product of Prime Factors

• Objective: Write a number using a set of prime factors that multiply to yield the original number.
• Example:
  • For 12, the prime factorization is 2 x 2 x 3, not just 2 x 3.

Using Factor Trees

• Purpose: To find prime factors of more complicated numbers.
• Method:
  1. Write the number at the top of the page (e.g., 220).
  2. Split into a factor pair (e.g., 220 into 110 and 2).
  3. Circle prime numbers.
  4. Factor non-prime numbers further (e.g., 110 into 11 and 10).
  5. Repeat until all factors are prime.
• Example:
  • 220: Split into 2, 2, 5, 11.
  • Prime factorization: 2^2 x 5 x 11.

Consistency in Factorization

• Different paths in factorizing a number lead to the same prime factors.
• Example:
  • 220 can split as 10 and 22, or 110 and 2, but will still result in the same set of prime factors.

Additional Example

• 112: Factorization process:
  1. Start with 112.
  2. Split into 2 and 56.
  3. Continue splitting 56 into 2 and 28, then 28 into 2 and 14, and finally 14 into 2 and 7.
  4. Result: Prime factors are 2, 2, 2, 2, 7.
  5. Prime factorization: 2^4 x 7.

Prime Factorization

• Definition: Rewriting a number as a product of its prime factors.
• The process shown is known as prime factorization.

Conclusion

• Covered the definition of prime factors, how to find them using factor trees, and examples.
• Emphasis on the consistency of factorization methods.
• Prime factorization allows expressing numbers uniquely as a product of primes.