Math Class: Prime Factorisation and HCF/LCM

Prime Factorisation

• Every whole number greater than 1 can be written as a product of prime numbers
• Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves
• To write a whole number as a product of primes, divide the number by the smallest prime number that divides it, and repeat the process until you reach a prime number
• For example, 24 can be written as 2 x 2 x 2 x 3, or 2³ x 3

Highest Common Factor (HCF)

• The HCF of two or more numbers is the highest number that divides into all of them without leaving a remainder
• To find the HCF, write each number as a product of prime numbers, then multiply all the common prime factors together
• For example, the HCF of 12 and 18 is 6, because 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3

Lowest Common Multiple (LCM)

• The LCM of two or more numbers is the lowest number that is a multiple of all of them
• To find the LCM, write each number as a product of prime numbers, then multiply together the highest powers of all prime factors
• For example, the LCM of 12 and 18 is 36, because 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3

Order of Operations (BIDMAS)

• BIDMAS stands for Brackets, Indices, Division and Multiplication, Addition and Subtraction
• This is the order in which operations should be performed in a mathematical expression
• For example, in the expression 2 + 3 x 4, the multiplication should be performed before the addition, so the answer is 14, not 20

Indices

• An index is a number that indicates how many times a base number is multiplied by itself
• For example, 2⁴ means 2 x 2 x 2 x 2 = 16
• Index rules include:
  • xᵐ x xⁿ = xᵐ⁺ⁿ
  • xᵐ ÷ xⁿ = xᵐ⁻ⁿ
  • (xᵐ)ⁿ = xᵐⁿ

Expressions as Powers

• Any expression can be written as a power of one number using index rules
• For example, x²y³ can be written as (xy)⁵

Forming Equations

• To form an equation from a word problem, identify the unknown quantity and represent it with a variable
• Then, write down the relationships between the quantities in the problem using mathematical symbols
• For example, if the problem says "The sum of two numbers is 10", you can write this as x + y = 10, where x and y represent the two numbers

Solving Linear Equations

• A linear equation is an equation that can be written in the form ax + b = 0, where a and b are constants and x is a variable
• To solve a linear equation, isolate the variable x on one side of the equation using inverse operations
• For example, to solve the equation 2x + 3 = 7, we can subtract 3 from both sides to get 2x = 4, and then divide both sides by 2 to get x = 2

Simplifying Expressions

• An expression is a combination of numbers and variables connected by mathematical operations
• To simplify an expression, combine like terms and use index rules
• For example, the expression 3x + 2y - x + 5y can be simplified to 2x + 7y

Substitution

• Substitution is the process of replacing a variable in an expression or equation with its value
• For example, if x = 2, then the expression 3x + 1 can be simplified by substituting 2 for x, giving 3(2) + 1 = 7

Factorising

• Factorising is the process of writing an expression as a product of its factors
• To factorise an expression, look for common factors and use the distributive property
• For example, the expression 2x² + 4x can be factorised by taking out a common factor of 2x, giving 2x(x + 2)

Prime Factorization

• Expressing a number as a product of prime numbers.
• A prime number is a number only divisible by 1 and itself.

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

• HCF is the highest number that divides two or more numbers.
• LCM is the smallest number that is a multiple of two or more numbers.

BIDMAS

• An order of operations for calculations, standing for:
  • Brackets
  • Indices
  • Division and Multiplication (from Left to Right)
  • Addition and Subtraction (from Left to Right)

Indices

• Indices show how many times a number is multiplied by itself.
• Simplify expressions using index rules:
  • a^m x aⁿ = a^m+n
  • a^m / aⁿ = a^m-n
  • (a^m)ⁿ = a^{m x n}

Expressions as a Power of One Number

• Rewrite expressions with repeated multiplications using index notation.

Forming Equations

• Translate written problems into mathematical equations, using variables to represent unknown quantities.

Solving Linear Equations

• Finding the value of the unknown variable in an equation.
• Use inverse operations to isolate the variable.

Simplifying Expressions

• Combine like terms by adding or subtracting their coefficients.
• Apply the rules of indices and other algebraic operations.

Substitution

• Replace variables in an expression or equation with their given values.

Factorising

• Expressing an expression as a product of its factors.
• Finding common factors to simplify expressions.