Multiplication and Division Basics

Definition: Multiplication is a mathematical operation representing the repeated addition of a number.
Notation: Commonly denoted by symbols such as ×, *, or dot (·).
Properties:
- Commutative Property: a × b = b × a
- Associative Property: (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = (a × b) + (a × c)
Multiplication Table: A structured table that lists the products of pairs of numbers.
Applications: Used in various fields such as algebra, geometry, and everyday calculations.

Definition: Division is the operation of distributing a number into equal parts or groups.
Notation: Commonly denoted by symbols such as ÷, /, or a fraction line (a/b).
Properties:
- Not Commutative: a ÷ b ≠ b ÷ a (only equal if a = b)
- Not Associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
- Distributive Over Addition: a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Inverse Operation: Division is the inverse operation of multiplication.
Common Terms:
- Dividend: The number being divided.
- Divisor: The number by which the dividend is divided.
- Quotient: The result of the division.
- Remainder: The amount left over when the dividend is not evenly divisible by the divisor.
Applications: Used in everyday scenarios, problem-solving, and in various fields including finance and science.

Definition: Represents repeated addition of a number, forming a fundamental arithmetic concept.
Notation: Symbols used include × (times), * (asterisk), and · (dot).
Commutative Property: Order of factors does not affect the product, i.e., a × b equals b × a.
Associative Property: Grouping of factors does not impact the product, i.e., (a × b) × c equals a × (b × c).
Distributive Property: Allows multiplication over addition, expressed as a × (b + c) equals (a × b) + (a × c).
Multiplication Table: A systematic layout showing products of pairs of numbers to facilitate quick reference.
Applications: Integral to various domains such as algebra, geometry, engineering, and everyday problem-solving.

Definition: Involves distributing a number into equal parts or groups, forming a basic mathematical operation.
Notation: Represented by symbols like ÷ (division sign), / (slash), or as a fraction line (a/b).
Not Commutative: Dividing numbers does not yield the same result if the order is reversed; a ÷ b is generally not equal to b ÷ a unless the numbers are equal.
Not Associative: Changing the grouping of the numbers affects the result, i.e., (a ÷ b) ÷ c does not equal a ÷ (b ÷ c).
Distributive Over Addition: Division does not distribute over addition, so a ÷ (b + c) cannot be simplified to (a ÷ b) + (a ÷ c).
Inverse Operation: Division is the opposite of multiplication, used to determine how many times a divisor fits into a dividend.
Common Terms:
- Dividend: The number subject to division.
- Divisor: The number that divides the dividend.
- Quotient: The result obtained from the division.
- Remainder: The leftover part when the dividend cannot be divided evenly by the divisor.
Applications: Crucial in daily tasks, mathematical problem-solving, finance, and scientific calculations.

Definition: A mathematical operation that involves adding a number to itself multiple times.
Notation: Commonly symbolized as '×' or '*'.
Properties:
- Commutative Property: The order of factors does not affect the product (e.g., a × b = b × a).
- Associative Property: The way factors are grouped does not change the product (e.g., (a × b) × c = a × (b × c)).
- Distributive Property: Multiplication distributes over addition (e.g., a × (b + c) = (a × b) + (a × c)).
Multiplication Table: A grid illustrating products of pairs of numbers, typically ranging from 1 to 10 or 12.

Definition: A mathematical operation that calculates how many times one number is contained in another.
Notation: Represented as '÷' or '/'.
Properties:
- Non-Commutative: The order of numbers matters (e.g., a ÷ b ≠ b ÷ a).
- Non-Associative: Grouping of numbers affects the outcome (e.g., (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)).
Long Division: A step-by-step method for dividing larger numbers, focusing on breaking the problem down into easier parts.
Remainders: The amount left over when one number cannot be evenly divided by another, important for understanding division results.

Definition: The largest integer that divides two or more numbers without leaving a remainder.
Methods to Find the GCF:
- Listing Factors: Identify all divisors of each number to find the largest common divisor.
- Prime Factorization: Decompose numbers into their prime factors and determine common factors.
- Euclidean Algorithm:
  - Divide the larger number by the smaller one.
  - Use the obtained remainder for the next division step.
  - Continue until the remainder is zero; the last non-zero remainder is the GCF.
Applications: Important for simplifying fractions, computing common denominators, and solving ratio-related problems.