Prime Numbers, Composite Numbers, and Prime Factorization Techniques

Classification of Natural Numbers

• Categorization of Numbers: According to the instructor, there are two primary types of natural numbers based on their divisibility.

• Prime Numbers: A natural number is defined as prime if its only factors are the number one and the number itself.

• Composite Numbers: Natural numbers that are not prime (meaning they have biological factors other than just one and themselves) are called composite numbers.

• Status of the Number One: The number $1$ is a unique exception in this classification system; it is categorized as neither prime nor composite.

• Learning Objective: The goal of this section is to develop skills in factoring composite numbers (phonetically referred to as "deposit numbers" in the session) and performing general factoring.

Foundational Examples of Prime Numbers

• The following is a list of several initial prime numbers where the only factors are the number itself and $1$:   - $2$   - $3$   - $5$   - $7$   - $11$   - $13$

Components of Multiplication and Factoring

• Definitions in Context: Multiplication and factoring are inverse operations involving specific components:   - Factors: The numbers that are multiplied together. For example, in the equation $3 \times 5 = 15$, the numbers $3$ and $5$ are the factors.   - Product: The result of the multiplication. In the same example, $15$ is the product.

• Operational Directionality:   - Standard Multiplication: The process involves moving from prime factors to a singular product.   - Factoring: The process involves starting with a product and decomposing it back into its prime factors. This is the ultimate objective of the lesson.

The Factor Tree Technique

• Methodology: The factor tree is identified as the easiest initial method for breaking down a number into its prime components.

• Procedure:   - Write down the composite number that needs to be factored.   - Break it down into the product of any two numbers that result in that original number.   - Algebraic Notation: In algebra, a dot ( $\cdot$ ) or standard multiplication symbol ( $\times$ ) is typically used to represent multiplication between factors.   - Continued Evaluation: Examine each factor to determine if it is prime. If a factor is not prime (composite), it must be factored further until every branch of the "tree" terminates in a prime number.

Detailed Example Walkthrough: Prime Factorization of $80$

• Initial Selection: The instructor initially mentioned finding the prime factorization of $24$ but pivoted to use the number $80$ for the demonstration.

• Step 1: Identify the first set of factors. The instructor chose $40$ and $2$ ($40 \times 2$).

• Step 2: Verify primality of factors.   - The factor $2$ is prime.   - The factor $40$ is not prime (composite) and requires further breakdown.

• Step 3: Factor the composite number $40$.   - The instructor mentions possibilities such as $10 \times 4$ or $5 \times 8$.   - Selecting $4 \times 10$, the expansion of $80$ is now expressed as $4 \times 10 \times 2$.

• Step 4: Verify primality of new factors.   - The factor $4$ is not prime; it can be factored into $2 \times 2$.   - The factor $10$ is not prime; it can be factored into $2 \times 5$.   - The final factor of $2$ is brought down.

• Step 5: Consolidate all branches. The complete list of factors is now:   - $2 \times 2 \times 2 \times 5 \times 2$

• Step 6: Final Result Assembly. By counting the occurrences, we find there are four $2$s and one $5$.   - Prime Factorization of $80$: $2 \times 2 \times 2 \times 2 \times 5$

Questions & Discussion

• Mathematical Notation and Exponents:   - Question: A student inquired if the final answer for the prime factorization of $80$ could be written in a more compact form using exponents, specifically as $2^4 \times 5$.   - Response: The instructor confirmed that writing the expression as $2$ raised to the fourth power times $5$ ($2^4 \times 5$) is mathematically correct. However, the instructor noted that they were "getting a little bit ahead of ourselves" since exponential notation had not been officially introduced in the curriculum yet at that point in the lesson.

Natural numbers are special numbers we use for counting things. There are two types:

• Prime Numbers: These are special numbers that can only be made by multiplying the number 1 and itself. So, for example, the number 2 can only be made by 1 × 2.

• Composite Numbers: These numbers have more friends, meaning you can make them by multiplying different numbers together. For example, the number 4 can be made by 2 × 2.

• The number 1 is different. It doesn’t fit into either group!

When we want to find out which special numbers make up a bigger number, we can use something called "factor trees." Imagine a tree that shows how a number breaks down into smaller numbers. We start with a big number, like 80, and split it into smaller parts, like 40 and 2, and keep going until we find the special numbers.

In the end, we can show the number 80 as 2 × 2 × 2 × 2 × 5. This means we used four 2's and one 5 to make 80!