Overview
This lecture explains prime factor decomposition, also known as writing a number as a product of its prime factors, using factor trees and index notation.
Key Terms: Factors and Primes
- A factor divides another number exactly without leaving a remainder.
- Prime numbers are positive integers with exactly two factors: 1 and itself.
- The first few prime numbers to memorize: 2, 3, 5, 7, 11, 13.
Prime Factor Decomposition Process
- Prime factor decomposition means expressing a number as a multiplication of prime numbers.
- Use a factor tree to break a number into its factor pairs, circling primes and stopping at prime branches.
- Always use multiplication signs (not addition or commas) when writing the product of primes.
- The process can start with any factor pair; the result will be a unique product of primes.
Example: Decomposing 40
- Begin with 40; split as 2 × 20 (circle 2—prime).
- 20 is 2 × 10 (circle 2).
- 10 is 2 × 5 (circle both; both are prime).
- Write: 2 × 2 × 2 × 5 or, using index form, 2³ × 5.
Example: Decomposing 120 in Index Form
- Start with 120; 2 × 60 (circle 2).
- 60 is 2 × 30 (circle 2).
- 30 is 2 × 15 (circle 2).
- 15 is 3 × 5 (circle both).
- Write: 2 × 2 × 2 × 3 × 5, or in index form: 2³ × 3 × 5.
Square Numbers and Prime Factorization Challenge
- Given P = 2 × 3² × a, find a so that P is a square number.
- If a = 2, then P = 2² × 3² = (2 × 3)² = 6², which is a square.
- If a = 8 (which is 2³), then P = 2⁴ × 3² = (2² × 3)² = 12², also a square.
- The smallest possible values of a are 2 and 8.
Key Terms & Definitions
- Factor — A number that divides exactly into another number.
- Prime number — A positive integer with exactly two factors: 1 and itself.
- Product — The result of multiplying two or more numbers together.
- Index form — Writing repeated multiplication of the same number using powers (exponents).
Action Items / Next Steps
- Complete four problems from each list in the Prime Factors Worksheet Pack.
- Attempt the challenge question at the bottom of the worksheet.