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Explore properties of numbers, including factors, multiples, prime numbers, and divisibility rules.

Patterns in Numbers: Number Theory

Number theory explores the properties and relationships of integers, including factors, multiples, prime numbers, and divisibility rules. These foundational concepts help students recognize patterns in mathematics, simplify fractions, find common denominators, and develop the logical thinking needed for algebra and beyond.

Components of Number Theory

This section breaks down the key concepts in number theory:

  • Factors & Multiples: Factors are numbers that divide evenly into another number; multiples are the products of a number.
  • Prime & Composite Numbers: Primes have exactly two factors (1 and themselves); composites have more than two.
  • Divisibility Rules: Shortcuts to determine if a number is divisible by 2, 3, 5, 9, 10, and other common divisors.
  • Greatest Common Factor & Least Common Multiple: GCF is the largest shared factor; LCM is the smallest shared multiple.

Examples of Number Theory

Factors & Multiples Examples

  • Factors of 12: 1, 2, 3, 4, 6, 12 β€” all numbers that divide evenly into 12.
  • First five multiples of 7: 7, 14, 21, 28, 35 β€” found by multiplying 7 Γ— 1, 7 Γ— 2, 7 Γ— 3, and so on.
  • A teacher arranges 24 chairs into equal rows. The possible row sizes are the factors of 24: 1, 2, 3, 4, 6, 8, 12, or 24.

Prime & Composite Examples

  • The number 17 is prime because its only factors are 1 and 17.
  • The number 18 is composite because it has factors 1, 2, 3, 6, 9, and 18.
  • To check if 29 is prime, test divisibility by 2, 3, and 5 (primes up to √29 β‰ˆ 5.4). None divide evenly, so 29 is prime.

Divisibility Rules Examples

  • 84 is divisible by 2 (ends in 4), by 3 (8 + 4 = 12, which is divisible by 3), and by 4 (84 Γ· 4 = 21).
  • 135 is divisible by 5 (ends in 5) and by 9 (1 + 3 + 5 = 9, which is divisible by 9).
  • To quickly check if 276 is divisible by 3: add the digits 2 + 7 + 6 = 15, and 15 is divisible by 3, so yes.

GCF & LCM Examples

  • GCF of 12 and 18: Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. The greatest common factor is 6.
  • LCM of 4 and 6: Multiples of 4 are 4, 8, 12, 16... Multiples of 6 are 6, 12, 18... The least common multiple is 12.
  • To add 1/8 + 1/6, find the LCM of 8 and 6, which is 24, then rewrite as 3/24 + 4/24 = 7/24.