Multiples, Factors & Primes

Finding Factors Systematicially

• To find all factors of a number, list them in factor pairs. Start with $1 \times n$, then $2 \times (n/2)$, and continue until the pairs begin to repeat or meet in the middle.

Prime Factor Decomposition

• Use a Factor Tree to break a number down into its prime components. Continue branching out until every end-node is a prime number, then express the original number as a product of these primes using index notation (e.g., $2^2 \times 3 \times 5$).

HCF and LCM via Venn Diagrams

• Place the prime factors of two numbers into a Venn diagram. The Highest Common Factor (HCF) is the product of the numbers in the intersection, while the Lowest Common Multiple (LCM) is the product of all numbers appearing in both circles.

• Prime vs. Composite: A prime number has exactly two factors. A composite number has more than two factors. Note that 1 is neither prime nor composite because it only has one factor.

• HCF vs. LCM: The HCF is the largest number that divides into both values (useful for simplifying). The LCM is the smallest number that both values divide into (useful for finding common denominators).

• The '1' Rule: Always remember that 1 is NOT a prime number. This is a frequent trick question in exams; prime numbers must have exactly two distinct factors.

• Divisibility Check: When checking if a large number is prime, you only need to test divisibility by primes up to the square root of that number. If no prime less than $\sqrt{n}$ divides $n$, then $n$ is prime.

• Reasonableness Check: For HCF and LCM, remember that the HCF cannot be larger than the smallest number in the set, and the LCM cannot be smaller than the largest number in the set.

• Confusing Factors and Multiples: Students often mix these up. Remember: Factors are Few (finite), and Multiples are Many (infinite).

• Missing the Number 2: Many students assume all prime numbers are odd. Always remember that 2 is the only even prime number and the smallest prime number.

• Incomplete Factor Trees: Ensure that every branch of a factor tree ends in a prime number. If a branch ends in a composite number like 9, it must be broken down further into $3 \times 3$.