2.3 Number Theory: Factors, Multiples, Primes, GCF & LCM

2.3 Number Theory: Factors, Multiples, Primes, GCF & LCM

Primes, Composites & Prime Factorization

Number theory focuses on the properties and relationships of integers. Understanding prime numbers, composite numbers, factors, and multiples provides the bedrock for fraction arithmetic and algebraic reasoning.

Definitions

• Factor (Divisor): A whole number that divides another whole number evenly, with no remainder.
• Multiple: The product of a given whole number and any non-zero whole number.
• Prime Number: A whole number strictly greater than $1$ that has exactly two distinct positive factors: $1$ and itself.
  • The first ten prime numbers are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
  • Note: $2$ is the smallest prime number and the ONLY even prime number.
• Composite Number: A whole number greater than $1$ that has more than two distinct positive factors.
  • Examples: $4, 6, 8, 9, 10, 12, 14, 15$.

[!WARNING] The Special Cases 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 factors.

Prime Factorization & The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer greater than $1$ is either a prime number itself or can be represented as a unique product of prime numbers, up to the order of the factors.

Method 1: Factor Tree To find the prime factorization of $360$:

1. Branch $360$ into any two factors: $36 \times 10$.
2. Branch $36$ into $6 \times 6$ and $10$ into $2 \times 5$.
3. Branch $6$ into $2 \times 3$ and the other $6$ into $2 \times 3$.
4. Collect all prime ends: $2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5^1$.

Method 2: Ladder / Division Method Divide successively by the smallest prime numbers:

• $360 \div 2 = 180$
• $180 \div 2 = 90$
• $90 \div 2 = 45$
• $45 \div 3 = 15$
• $15 \div 3 = 5$
• $5 \div 5 = 1$ Prime factors: $2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5^1$.

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Divisibility Rules

Divisibility rules enable candidates to test quickly whether a number is divisible by another without performing full long division.

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Greatest Common Factor (GCF) & Least Common Multiple (LCM)

Definitions & Calculation Methods

• Greatest Common Factor (GCF): The largest whole number that divides evenly into two or more numbers.
• Least Common Multiple (LCM): The smallest positive whole number that is a multiple of two or more numbers.

Method: Prime Factorization (Exponential Comparison)

To find the GCF and LCM of $48$ and $72$:

1. Write prime factorizations: $48 = 2^4 \times 3^1$ $72 = 2^3 \times 3^2$
2. GCF Rule: Take the MINIMUM exponent of each shared prime factor. $\text{GCF}(48, 72) = 2^{\min(4,3)} \times 3^{\min(1,2)} = 2^3 \times 3^1 = 8 \times 3 = \mathbf{24}$
3. LCM Rule: Take the MAXIMUM exponent of every prime factor present. $\text{LCM}(48, 72) = 2^{\max(4,3)} \times 3^{\max(1,2)} = 2^4 \times 3^2 = 16 \times 9 = \mathbf{144}$

Useful Formula Relationship

For any two positive integers $a$ and $b$: $a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b)$ $\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}$

Verification for $48$ and $72$: $48 \times 72 = 3,456 \quad \text{and} \quad 24 \times 144 = 3,456 \quad \checkmark$

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Real-World Word Problems: GCF vs. LCM

A core skill tested on Praxis 5003 is determining whether a word problem requires finding the GCF or the LCM.

Decision Matrix for Word Problems

Step-by-Step Word Problem Examples

Worked Example 1 (GCF Context):

A florist has 60 red roses and 84 white tulips. He wants to create identical floral arrangements using all flowers with no flowers left over. What is the maximum number of arrangements he can create, and how many of each flower will be in each arrangement?

Solution:

1. Recognize problem type: Equal division without leftovers $\rightarrow$ GCF.
2. Prime factorizations: $60 = 2^2 \times 3^1 \times 5^1$ $84 = 2^2 \times 3^1 \times 7^1$
3. Find GCF: $\text{GCF}(60, 84) = 2^2 \times 3^1 = 4 \times 3 = \mathbf{12}$
4. Interpret result:
   • Maximum arrangements = $12$ arrangements.
   • Red roses per arrangement: $60 \div 12 = \mathbf{5}$ roses.
   • White tulips per arrangement: $84 \div 12 = \mathbf{7}$ tulips.

Worked Example 2 (LCM Context):

Red flashing lights blink every 15 seconds, and blue flashing lights blink every 20 seconds. If both lights blink simultaneously at 12:00 PM, at what time will they next blink together?

Solution:

1. Recognize problem type: Repeating periodic cycles lining up $\rightarrow$ LCM.
2. Prime factorizations: $15 = 3^1 \times 5^1$ $20 = 2^2 \times 5^1$
3. Find LCM: $\text{LCM}(15, 20) = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = \mathbf{60} \text{ seconds}$
4. Interpret result:
   • $60 \text{ seconds} = 1 \text{ minute}$.
   • The lights will next blink together at 12:01 PM.