Prime Factorization and Factor Trees

What are Prime Factors?

• Prime factors are factors of a number that are also prime numbers.
• Example: For the number 12, the factors are 1, 2, 3, 4, 6, and 12. Its prime factors are 2 and 3.

Expressing a Number as a Product of Prime Factors

• To express a number as a product of prime factors, find a set of prime factors that multiply to the number.
• Example: To express 12 as a product of its prime factors:
  • Use 2 and 3, but since 2 x 3 = 6, not 12, we use 2 x 2 x 3 = 12.

Factor Trees

• A method to find prime factors of a number, especially useful for larger numbers.
• Steps:
  1. Write the number at the top of the page.
  2. Begin factorizing by splitting the number into two factors.
  3. Circle any prime numbers and leave them alone.
  4. Continue factorizing non-prime numbers until all factors are prime.

Example 1: Prime Factorization of 220

• Start with 220.
• Split into 110 and 2 (circle 2 as it is prime).
• Split 110 into 11 and 10 (circle 11 as it is prime).
• Split 10 into 5 and 2 (circle both as they are prime).
• Prime factors are 2, 2, 5, and 11.
• Express as: 2² x 5 x 11.
• Note: Different splitting methods will still result in the same prime factors.

Example 2: Prime Factorization of 112

• Start with 112.
• Split into 2 and 56 (circle 2 as it is prime).
• Split 56 into 2 and 28 (circle 2).
• Split 28 into 2 and 14 (circle 2).
• Split 14 into 2 and 7 (both are prime, circle them).
• Prime factors are 2, 2, 2, 2, and 7.
• Express as: 2⁴ x 7.

Prime Factorization

• The process of rewriting a number as a product of its prime factors.
• Example: The prime factorization of 112 is 2⁴ x 7.

Conclusion

• Prime factorization is a useful skill in mathematics, especially for exams.
• Factor trees simplify the process of finding prime factors.
• Different methods of factorizing will still result in the same prime factors, ensuring consistency.