2.1 Factors, Prime Factors and Highest Common Factor (HCF)
1. Factors
Factors of a whole number are whole numbers that can divide that number completely without leaving any remainder.
- Example: The factors of 12 are 1, 2, 3, 4, 6, and 12 (since $12 \div 1 = 12$, $12 \div 2 = 6$, $12 \div 3 = 4$).
2. Prime Factors
Prime factors are factors of a number that are also prime numbers (numbers greater than 1 that have only two factors: 1 and themselves).
- Prime Factorisation Methods:
- Factor Tree Method: Repeatedly break down a number into pairs of factors until all branch ends are prime.
- Repeated Division Method: Divide the number continuously by its smallest prime factors until the quotient is 1.
- Example: Prime factorisation of $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$.
3. Common Factors and Highest Common Factor (HCF)
Common factors are numbers that are factors of two or more given numbers.
The Highest Common Factor (HCF) is the largest common factor among those numbers.
- Methods to find HCF:
- Listing Common Factors: List all factors for each number and pick the largest shared factor.
- Repeated Division: Divide the numbers concurrently by their common prime factors until there are no more common factors left.
- Prime Factorisation: Find the product of the common prime factors with the smallest powers.
2.2 Multiples, Common Multiples and Lowest Common Multiple (LCM)
1. Multiples and Common Multiples
A multiple of a number is the product of that number and any non-zero whole number.
Common multiples are numbers that are multiples of two or more given numbers.
- Example: Multiples of 3 are 3, 6, 9, 12, 15... | Multiples of 4 are 4, 8, 12, 16... Common multiples of 3 and 4 include 12, 24, 36...
2. Lowest Common Multiple (LCM)
The Lowest Common Multiple (LCM) is the smallest positive common multiple among two or more given numbers.
- Methods to find LCM:
- Listing Multiples: List the multiples of each number until you find the first common value.
- Repeated Division: Divide all numbers by prime factors continuously until all quotients become 1.
- Prime Factorisation: Express each number as a product of prime factors, then select each prime factor with its highest power.
2.3 Problem Solving Involving HCF and LCM
When solving real-world word problems, identify whether to use HCF or LCM based on key indicators:
- Use HCF when: You need to divide or split items into equal groups, equal lengths, or equal sets of maximum size without any leftover (e.g., maximum size of tiles, equal Distribution of items).
- Use LCM when: Events occur repeatedly at regular intervals and you need to find when they will happen together or simultaneously again (e.g., traffic lights blinking together, alarm clocks ringing at the same time).