Factors and Multiples

• Mathematics involves the study of numbers, quantities, shapes, and patterns
• Number theory is a branch of mathematics that focuses on the properties and relationships of numbers, especially integers

Factors

• Factors of a number are integers that divide the number evenly without leaving a remainder
• 1, 2, 3, 4, 6, and 12 are factors of 12 because each of these numbers divides 12 without leaving a remainder
• Both 1 and the number itself are always factors of the number
• Prime numbers have exactly two distinct factors: 1 and the number itself
• Finding factors involves checking which numbers divide the given number evenly
• Factorization is the process of expressing a number as a product of its factors
• Prime factorization is expressing a number as a product of its prime factors
• The prime factorization of 24 is 2 x 2 x 2 x 3, or 2^3 x 3

Multiples

• A multiple of a number is the product of that number and any integer
• 5, 10, 15, 20, and 25 are multiples of 5 because they are the result of multiplying 5 by an integer
• To find multiples of a number, multiply the number by successive integers (1, 2, 3,...)
• The least common multiple (LCM) of two or more numbers is the smallest multiple that is common to all of them
• The LCM of 4 and 6 is 12 because 12 is the smallest number that is a multiple of both 4 and 6
• Multiples are infinite for any given number since you can always multiply by a larger integer

Multiplication

• Multiplication is a basic arithmetic operation that represents repeated addition of a number
• It is the process of finding the product of two or more numbers
• The numbers being multiplied are called factors or multipliers, and the result is called the product
• Multiplication is commutative, meaning the order of the factors does not affect the product (a x b = b x a)
• Multiplication is associative, meaning the grouping of factors does not affect the product (a x (b x c) = (a x b) x c)
• The identity element for multiplication is 1, because any number multiplied by 1 equals itself (a x 1 = a)
• The zero property of multiplication states that any number multiplied by 0 equals 0 (a x 0 = 0)

Division

• Division is the inverse operation of multiplication, used to split a number into equal parts
• It involves finding how many times one number (the divisor) is contained in another number (the dividend)
• The result of division is called the quotient; if there is a remainder, it is the amount left over
• Division by zero is undefined because it leads to logical inconsistencies
• The division algorithm states that for any two integers a (dividend) and b (divisor, not zero), there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < |b|
• Division can be represented as a fraction, where the dividend is the numerator and the divisor is the denominator

Percentage

• A percentage is a way of expressing a number as a fraction of 100
• The term "percent" means "per hundred" or "out of 100"
• Percentages are used to express proportions and rates
• To convert a fraction or decimal to a percentage, multiply it by 100 and add the percent sign (%)
• 0. 75 = 0.75 x 100 = 75%
• To convert a percentage to a decimal, divide it by 100
• 40% = 40 / 100 = 0.40
• To find a percentage of a number, multiply the number by the percentage expressed as a decimal
• To find 20% of 50, calculate 0.20 x 50 = 10
• Percentage change is the extent to which a quantity gains or loses value
• It is calculated as [(New Value - Old Value) / Old Value] x 100
• Percentage increase occurs when the new value is greater than the old value, resulting in a positive percentage change
• Percentage decrease occurs when the new value is less than the old value, resulting in a negative percentage change