Comprehensive Guide to Prime Factorization and Factor Trees

Definition of Factors: Factors are defined as two or more numbers that, when multiplied together, form a product.
Example 1: Factors of $6$
- Equations: $2 \cdot 3 = 6$ and $1 \cdot 6 = 6$ .
- The factors of $6$ are $1$ , $6$ , $2$ , and $3$ .
- Reasoning: When these numbers are divided into $6$ , there is no remainder.
- Non-factor: $5$ would NOT be a factor of $6$ because $5$ does not evenly divide into $6$ .
Example 2: Factors of $12$
- Equations: $3 \cdot 4 = 12$ and $2 \cdot 6 = 12$ .
- The factors of $12$ are $3$ , $4$ , $2$ , and $6$ .
- Reasoning: When these numbers are divided into $12$ , there is no remainder.
- Non-factor: $5$ would NOT be a factor of $12$ because $5$ does not evenly divide into $12$ .

Prime Numbers
- Definition: A prime number is a whole number that has exactly $2$ unique factors: $1$ and itself.
- Examples: $2$ , $3$ , $23$ , $37$ , $41$ .
- Special Property: The number $2$ is the only prime number that is even.
Composite Numbers
- Definition: A composite number is a whole number that has more than $2$ factors.
- Examples: $15$ , $410$ , $100$ , $91$ .
Special Cases (Neither Prime nor Composite)
- The numbers $0$ and $1$ are excluded from both categories; they are neither prime nor composite.

Definition: Prime factorization is the process of writing a composite number as the product of prime factors.
The Factor Tree Method: This is a visual tool used to find the prime factorization of a number.
- Step 1: Write the number you are factoring at the top of the tree.
  - Example: Write $20$ at the top.
- Step 2: Choose any pair of whole numbers that are factors of the target number.
  - Example: For $20$ , choose factors $4$ and $5$ .
- Step 3: Continue to factor any number that is not prime.
  - Example: Since $5$ is prime, it is left alone. Since $4$ is composite, it is factored into $2 \cdot 2$ .
- Completion: The factor tree is complete only when you are left with nothing but prime numbers.

Efficiency Tip: Circle the prime numbers as you identify them throughout the process. This ensures you can easily see all prime factors when the tree is finished.
Formatting the Final Answer:
- Order: Prime factors in the final answer should be written in order from least to greatest.
- Multiplication Symbol: Do not use the variable $x$ as the multiplication symbol.
- Form: Write the final answer as a product of the prime factors identified.
Comprehensive Example: Prime Factorization of $90$
- Regardless of which initial factors you choose, the outcome remains the same.
- Path A: Start with factors $9$ and $10$ .
  - Factor $9$ into $3 \cdot 3$ .
  - Factor $10$ into $2 \cdot 5$ .
- Path B: Start with factors $30$ and $3$ .
  - Factor $30$ into $10 \cdot 3$ .
  - Factor $10$ into $2 \cdot 5$ .
- Final Factors Identified: $2$ , $3$ , $3$ , and $5$ .
- Final Answer: $2 \cdot 3 \cdot 3 \cdot 5$