Number Theory and Operations

TL;DR

Number theory explores properties of integers, like primes and divisibility, while operations are the basic arithmetic actions you perform on them. Understanding these helps you solve problems involving quantities and relationships between numbers. You'll learn how to break numbers down, build them up, and use them efficiently.

1. The Mental Model

Think of numbers as building blocks. Number theory is like understanding what kinds of blocks exist (e.g., small, unbreakable primes) and how they fit together. Operations are the tools you use to connect or separate these blocks.

2. The Core Material

Number theory is all about the integers (whole numbers: ..., -2, -1, 0, 1, 2, ...). We'll focus on positive integers for most of this.

Divisibility and Factors

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A number 'a' is divisible by 'b' if dividing 'a' by 'b' leaves no remainder. In this case, 'b' is a factor (or divisor) of 'a', and 'a' is a multiple of 'b'.

• Example: 12 is divisible by 3 because 12 ÷ 3 = 4 (no remainder). So, 3 is a factor of 12, and 12 is a multiple of 3.

Prime and Composite Numbers

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• A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. (Examples: 2, 3, 5, 7, 11)
• A composite number is a natural number greater than 1 that is not prime; it has more than two distinct positive divisors. (Examples: 4, 6, 8, 9, 10)
• The number 1 is neither prime nor composite.

Prime Factorization

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Every composite number can be expressed as a unique product of prime numbers. This is called prime factorization.

• Example:
  • 12 = 2 × 2 × 3 = 2² × 3
  • 30 = 2 × 3 × 5

graph TD
    A["Number"] --> B{"Is it prime?"}
    B -- "Yes" --> C["Done (The number itself is its prime factor)"]
    B -- "No" --> D["Find smallest prime factor (p)"]
    D --> E["Divide Number by p (New Number)"]
    E --> F["Add p to list of prime factors"]
    F --> A

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

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• GCD (Greatest Common Divisor): The largest number that divides into two or more numbers without leaving a remainder.

  • To find GCD: List prime factors for each number, then multiply the common prime factors (using the lowest power).
  • Example: GCD(12, 18)
    • 12 = 2² × 3
    • 18 = 2 × 3²
    • Common factors: 2¹ and 3¹. So, GCD(12, 18) = 2 × 3 = 6.

• LCM (Least Common Multiple): The smallest positive number that is a multiple of two or more numbers.

  • To find LCM: List prime factors for each number, then multiply the highest power of all prime factors that appear in any of the factorizations.
  • Example: LCM(12, 18)
    • 12 = 2² × 3¹
    • 18 = 2¹ × 3²
    • Highest powers: 2² and 3². So, LCM(12, 18) = 2² × 3² = 4 × 9 = 36.

Basic Operations Review

You already know these, but it's good to remember their properties:

• Addition (+): Combining quantities. (Commutative: a+b = b+a; Associative: (a+b)+c = a+(b+c))
• Subtraction (-): Finding the difference. (Not commutative or associative)
• Multiplication (× or *): Repeated addition. (Commutative: a×b = b×a; Associative: (a×b)×c = a×(b×c); Distributive: a×(b+c) = a×b + a×c)
• Division (÷ or /): Splitting into equal parts. (Not commutative or associative; division by zero is undefined)

3. Worked Example

Let's find the GCD and LCM of 24 and 60.

1. Prime Factorization:

   • For 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3¹
   • For 60: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

2. GCD (Greatest Common Divisor):

   • Identify common prime factors: 2 and 3.
   • Take the lowest power for each common factor: 2² (from 24 and 60) and 3¹ (from 24 and 60).
   • GCD(24, 60) = 2² × 3¹ = 4 × 3 = 12

3. LCM (Least Common Multiple):

   • Identify all prime factors that appear in either factorization: 2, 3, and 5.
   • Take the highest power for each factor: 2³ (from 24), 3¹ (from both), and 5¹ (from 60).
   • LCM(24, 60) = 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120

4. Key Takeaways

• Divisibility means one number can be divided by another with no remainder.
• Prime numbers are only divisible by 1 and themselves; all other whole numbers greater than 1 are composite.
• Prime factorization breaks down any composite number into a unique set of prime numbers multiplied together.
• The GCD is the largest number that divides into all given numbers.
• The LCM is the smallest number that is a multiple of all given numbers.

Common Mistakes to Avoid

• Forgetting that 1 is neither prime nor composite.
• Confusing GCD and LCM, especially when choosing powers of prime factors.
• Not finding all prime factors before calculating GCD or LCM.
• Dividing by non-prime numbers when doing prime factorization (e.g., saying 12 = 4 × 3 instead of 2 × 2 × 3).

5. Now Try It

Find the prime factorization, GCD, and LCM for the numbers 36 and 48. Write down each step clearly. You'll know you've succeeded if your prime factorizations are correct and your GCD and LCM values are 12 and 144, respectively.