2.3 Factors, Multiples, Divisibility Rules, & Prime Factorization

2.3 Factors, Multiples, Divisibility Rules, & Prime Factorization

Number theory concepts—including prime factorization, divisibility rules, Greatest Common Factor (GCF), and Least Common Multiple (LCM)—are heavily tested on TABE Mathematics Levels M, D, and A. These principles are vital for simplifying algebraic fractions, finding common denominators, and solving real-world scheduling and resource distribution problems.

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Factors, Multiples, & Number Classification

Definitions

• Factor (Divisor): An integer that divides evenly into another integer without leaving a remainder. For example, the factors of $24$ are ${1, 2, 3, 4, 6, 8, 12, 24}$.
• Multiple: The product of a given number and any integer. The positive multiples of $6$ are ${6, 12, 18, 24, 30, 36, \dots}$.

$\text{If } a \times b = c, \text{ then } a \text{ and } b \text{ are factors of } c, \text{ and } c \text{ is a multiple of both } a \text{ and } b.$

Prime vs. Composite Numbers

• Prime Number: A positive integer strictly greater than $1$ that has exactly two distinct positive divisors: $1$ and itself.
  • The first ten prime numbers are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
  • The number $2$ is the smallest prime and the only even prime number.
• Composite Number: A positive integer strictly greater than $1$ that has more than two distinct positive divisors (e.g., $4, 6, 8, 9, 10, 12$).

[!CAUTION] The Special Status of 0 and 1:

• The number $1$ is NEITHER prime NOR composite because it has only one positive factor ($1$).
• The number $0$ is NEITHER prime NOR composite because it has infinitely many divisors, but division by zero is undefined.

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Master Divisibility Rules (2 through 10)

Divisibility rules allow you to test whether large numbers can be divided evenly without carrying out full long division.

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Prime Factorization Algorithms

The Fundamental Theorem of Arithmetic states that every composite integer greater than $1$ can be expressed as a unique product of prime numbers (up to the order of the factors).

Method 1: Factor Tree

Continually split composite factors into factor pairs until every branch ends in a prime number.

         180
        /   \
      18     10
     /  \   /  \
    2    9 2    5
        / \
       3   3

Collecting prime terminal nodes: $180 = 2 \times 2 \times 3 \times 3 \times 5 = \mathbf{2^2 \times 3^2 \times 5^1}$.

Method 2: Upside-Down Division Ladder

Repeatedly divide the number by the smallest possible prime divisor until the quotient is $1$.

2 & 180 \\ \hline 2 & 90 \\ \hline 3 & 45 \\ \hline 3 & 15 \\ \hline 5 & 5 \\ \hline & 1 \end{array} \implies 180 = \mathbf{2^2 \times 3^2 \times 5^1}$$ --- ## Greatest Common Factor (GCF) vs. Least Common Multiple (LCM) | Attribute | Greatest Common Factor (GCF) | Least Common Multiple (LCM) | | :--- | :--- | :--- | | **Definition** | Largest factor dividing all given numbers evenly. | Smallest positive multiple shared by all given numbers. | | **Prime Factor Rule** | Take the **lowest exponent** of only the **shared** prime factors. | Take the **highest exponent** of **every** prime factor present. | | **Magnitude** | $\text{GCF} \le \text{smallest number}$ | $\text{LCM} \ge \text{largest number}$ | | **Formula** | $\text{GCF}(a,b) = \frac{a \times b}{\text{LCM}(a,b)}$ | $\text{LCM}(a,b) = \frac{a \times b}{\text{GCF}(a,b)}$ | ### Step-by-Step Calculation: GCF and LCM of $72$ and $108$ 1. Find prime factorizations: - $72 = 2^3 \times 3^2$ - $108 = 2^2 \times 3^3$ 2. **Compute GCF (Lowest Shared Exponents):** $$\text{GCF} = 2^{\min(3,2)} \times 3^{\min(2,3)} = 2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$$ 3. **Compute LCM (Highest All Exponents):** $$\text{LCM} = 2^{\max(3,2)} \times 3^{\max(2,3)} = 2^3 \times 3^3 = 8 \times 27 = \mathbf{216}$$ 4. **Verify Fundamental Product Relationship:** $$\text{GCF} \times \text{LCM} = 36 \times 216 = 7,776$$ $$a \times b = 72 \times 108 = 7,776 \quad \checkmark$$ --- ## Word Problem Discrimination: GCF vs. LCM Scenarios A critical skill on the TABE exam is recognizing whether a word problem requires finding the GCF or the LCM. ### GCF Clues & Scenarios (Dividing / Partitioning) - **Keywords:** *"Greatest number of identical kits"*, *"divided equally without leftovers"*, *"maximum length of equal strips"*, *"arranged in equal rows"*. - **Problem Structure:** You have fixed quantities of multiple items and want to split or group them into the largest possible identical sets. - **Example:** A teacher has $48$ pencils and $64$ erasers. What is the greatest number of identical prize packs she can create with no supplies left over? $$\text{GCF}(48, 64) = 16 \text{ prize packs (each with } 3 \text{ pencils and } 4 \text{ erasers)}$$ ### LCM Clues & Scenarios (Repeating Cycles / Synchronization) - **Keywords:** *"When will they next happen at the same time?"*, *"smallest number of packages to have equal items"*, *"beeps every X minutes and Y minutes"*, *"revolving cycles"*. - **Problem Structure:** Two or more repeating intervals start together, and you must find when they coincide again. - **Example:** Bus Route A arrives every $15$ minutes; Bus Route B arrives every $20$ minutes. If both buses depart at 8:00 AM, at what time will they next depart simultaneously? $$\text{LCM}(15, 20) = 60 \text{ minutes} \implies 60 \text{ minutes after 8:00 AM} = \mathbf{\text{9:00 AM}}$$