Theory of Positive Integers

An exploration of the properties, relationships, and applications of positive integers in various mathematical contexts.

Introduction to Positive IntegersDivisibility and MultiplesPrime Numbers and Composite Numbers
Greatest Common Divisor and Least Common MultipleThe Fundamental Theorem of ArithmeticNumber Patterns and Sequences
Using Positive Integers in Everyday LifePositive Integers in TechnologyOrganizing Events and Schedules
Breaking Down Problems with FactorizationUsing Patterns to Predict and Solve
Introduction to Posi...Divisibility and Mul...Prime Numbers and Co...Greatest Common Divi...The Fundamental Theo...Number Patterns and ...Using Positive Integ...Positive Integers in...Organizing Events an...Breaking Down Proble...Using Patterns to Pr...
Advanced Topics

Greatest Common Divisor and Least Common Multiple

The Greatest Common Divisor (GCD)

The GCD of two numbers is the largest positive integer that divides both numbers exactly. To find it, list the factors of each number and find the biggest one they share.

The Least Common Multiple (LCM)

The LCM is the smallest positive integer that is a multiple of both numbers. It's useful for finding when things "line up," like when two clocks chime together.

How to Find Them

  • GCD: List all factors, pick the biggest shared one.
  • LCM: List multiples, pick the smallest shared one.

Why Care?

These concepts help with simplifying fractions, scheduling, and solving puzzles!

Quick Formula

\[ \text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b \]

Key Formula

\[\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b\]

Examples

  • GCD of 12 and 18 is 6; LCM is 36.

  • GCD of 8 and 14 is 2; LCM is 56.

In a Nutshell

GCD and LCM help us find the largest shared factors and the smallest shared multiples of numbers.

Key Terms

Divisor
A number that divides another number exactly.
Multiple
A number produced by multiplying a given number by an integer.
Next
The Fundamental Theorem of Arithmetic