Chapter focus
Common factors and common multiples reveal how numbers fit together. HCF helps form the largest equal groups; LCM finds the first shared point in repeating cycles.
These original notes explain the standard meanings of the chapter title. Check the prescribed Part II textbook for the exact lesson sequence and exercises.
1. Factors and multiples
A factor divides a number exactly. The factors of 18 are 1, 2, 3, 6, 9 and 18. A multiple is found by multiplying by a whole number: multiples of 18 include 18, 36, 54, 72, … .
Common factors divide each of the given numbers. Common multiples are divisible by each of them.
2. Find HCF and LCM
The highest common factor (HCF) is the greatest common factor. The least common multiple (LCM) is the smallest positive common multiple.
Worked example: 24 and 36
- Prime factorise: 24 = 23 × 3; 36 = 22 × 32.
- For the HCF, use common primes with the smaller exponents: 22 × 3 = 12.
- For the LCM, use each prime with the greater exponent: 23 × 32 = 72.
Check for two positive integers: HCF × LCM = product of the numbers. Here 12 × 72 = 24 × 36 = 864.
3. Choose HCF or LCM for a situation
Use HCF when dividing quantities into the largest equal groups or cutting equal pieces with no remainder. Use LCM when periodic events repeat and you need the earliest time they coincide.
Example: Red lights flash every 8 seconds and blue lights every 12 seconds. They flash together again after LCM(8, 12) = 24 seconds.
Common errors to avoid
- HCF is no greater than the smaller number; LCM is at least as large as the greater number.
- In prime factorisation, choose smaller powers for HCF and larger powers for LCM.
- Do not use HCF for a “when will cycles coincide?” question.
- Check that the result divides both numbers or is a multiple of both, as required.
Practice: find and apply
1. Find the HCF of 84 and 180.
Answer: 84 = 22 × 3 × 7; 180 = 22 × 32 × 5. HCF = 22 × 3 = 12.
2. Find the LCM of 6 and 15.
Answer: 6 = 2 × 3 and 15 = 3 × 5; LCM = 2 × 3 × 5 = 30.
3. Two bells ring every 9 and 12 minutes. When do they next ring together?
Answer: LCM(9, 12) = 36 minutes.
4. Can 24 counters be split into the greatest possible equal groups of 18 and 36 counters without leftovers? How many counters per group?
Answer: Use HCF(18, 36) = 18; 18 counters per group is the greatest equal size that divides both totals.
5. Find the HCF and LCM of 8 and 20, then check their product.
Answer: HCF = 4 and LCM = 40. Their product is 160, equal to 8 × 20.
Chapter summary
Factors and multiples describe how whole numbers divide and repeat. HCF finds the greatest shared divisor; LCM finds the first positive shared multiple. Choose the one that fits the situation and verify your result.