Common Factors and Multiples Revision Notes | GCSE AQA Higher Maths

1. Why Factors and Multiples Matter

You're planning a party and need to arrange 24 guests into equal groups. How many different ways can you do this? This everyday problem connects directly to factors - the numbers that divide exactly into 24.

Factors and multiples appear everywhere in mathematics and real life. From simplifying fractions to finding when buses arrive at the same stop, these concepts help us solve practical problems efficiently.

The key insight is that every number can be broken down into prime factors - like finding the DNA of a number. Once you master this technique, calculating HCF and LCM becomes straightforward.

2. Building Blocks - Prime Factorisation

Every positive integer can be written as a product of prime numbers in exactly one way. This is called the Fundamental Theorem of Arithmetic (also known as the unique factorisation theorem), and it's the foundation for finding factors and multiples.

Think of prime factorisation as breaking a number down to its simplest building blocks. Just like how water breaks down to hydrogen and oxygen, numbers break down to prime factors.

Worked example: Factor Tree for 60

Find the prime factorisation of 60

1. 60 — Write the number at the top
2. 60 = 6 × 10 — Choose any factor pair - doesn't matter which
3. 6 = 2 × 3 10 = 2 × 5 — Keep going until all factors are prime
4. 60 = 2 × 2 × 3 × 5 — Collect all the prime factors
5. 60 = 2² × 3 × 5 — Group repeated primes using indices

Answer: 60 = 2² × 3 × 5

3. Finding the Highest Common Factor (HCF)

The HCF is like finding the biggest tool that can measure all your lengths exactly. If you have ropes of 18m and 24m, the HCF tells you the longest piece you could use to measure both without any leftover.

There are two main methods for finding HCF: listing factors or using prime factorisation. Prime factorisation is more efficient for larger numbers and connects to the deeper structure of mathematics. When using prime factorisation, the calculations should be accurate, leading to HCF of 24 when working with 48 and 72.

Worked example: Finding HCF Using Prime Factorisation

Find the HCF of 48 and 72

1. 48 = 2⁴ × 3 72 = 2³ × 3² — Break each number into prime factors
2. Common primes: 2 and 3 — Both numbers contain 2 and 3 as factors
3. 2: min(4,3) = 3, so 2³ 3: min(1,2) = 1, so 3¹ — Use the smallest power of each common prime
4. HCF = 2³ × 3¹ = 8 × 3 = 24 — The HCF is the product of common prime factors

Answer: HCF(48, 72) = 24

4. Finding the Lowest Common Multiple (LCM)

The LCM helps solve timing problems. If one bus comes every 12 minutes and another every 18 minutes, the LCM tells you when they'll arrive together again.

For LCM using prime factorisation, we take the highest power of each prime that appears in any of the numbers. This ensures our result is divisible by all the original numbers. The correct calculation shows that the LCM of 48 and 72 is 144.

Worked example: Finding LCM Using Prime Factorisation

Find the LCM of 48 and 72

1. 48 = 2⁴ × 3¹ 72 = 2³ × 3² — Express each number as a product of primes
2. All primes present: 2 and 3 — List every prime that appears in either factorisation
3. 2: max(4,3) = 4, so 2⁴ 3: max(1,2) = 2, so 3² — Use the largest power of each prime factor
4. LCM = 2⁴ × 3² = 16 × 9 = 144 — The LCM is the product of all prime factors at highest powers

Answer: LCM(48, 72) = 144

5. Practice and Application

Now you understand the theory, let's apply these methods to build confidence. Remember that prime factorisation gives you both HCF and LCM efficiently from the same working. It's crucial to carefully check calculations to ensure accuracy.

Worked example: Real-World Application

Traffic lights change every 45 seconds on one road and every 60 seconds on another. If they both change at 3:00 PM, when will they next change together?

1. Find LCM(45, 60) — We need the smallest time when both cycles align
2. 45 = 3² × 5 60 = 2² × 3 × 5 — Break down to prime factors
3. LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 — Take highest power of each prime: 2², 3², 5¹
4. 180 seconds = 3 minutes — 180 ÷ 60 = 3 minutes
5. 3:00 PM + 3 minutes = 3:03 PM — They will change together again at 3:03 PM

Answer: The traffic lights will change together again at 3:03 PM

6. Exam Success and Common Traps

Examiners often test whether you truly understand the difference between HCF and LCM. Many students confuse which method to use for which problem type.

The key is recognising the context. HCF problems involve sharing, dividing, or finding common measures. LCM problems involve timing, repeating cycles, or finding when events coincide.

Another common area for marks is showing your working clearly. Even if you know the answer, you must demonstrate the method to earn full marks in GCSE mathematics.