Factors, multiples and prime building blocks
Learn to recognise equal-group problems and repeating-cycle problems, then choose HCF or LCM for a reason.
Core and Extended. Together with Number families and reciprocals, this completes the first teaching sequence for section 1.1.
Before you begin
- Use multiplication facts and exact division.
- Recognise a prime number.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which is a factor of 18?
What you will learn
- Build a prime factorisation and verify it.
- Use HCF for the greatest number of identical groups.
- Use LCM for the first shared repeat.
Factors fit inside; multiples repeat
A factor of a positive integer divides it exactly. Factor pairs help you find all factors: for 18 the pairs are 1 × 18, 2 × 9 and 3 × 6, giving 1, 2, 3, 6, 9, 18. Stop once the pairs begin to repeat. A multiple is obtained by multiplying by an integer: the positive multiples of 18 begin 18, 36, 54, 72.
The direction matters. Three is a factor of 18; 18 is a multiple of 3. For positive integers, the highest common factor cannot exceed the smaller number, and the lowest positive common multiple cannot be smaller than the larger number. These checks often catch a swapped method.
Break numbers into primes
A prime has exactly two positive factors. To factorise a number, split it into a product, then keep splitting any factor that is not prime. For 60, one route is 6 × 10 = 2 × 3 × 2 × 5 = 2² × 3 × 5. A route through 4 × 15 gives the same prime building blocks.
Only the prime leaves belong in the final product. Do not also multiply by the intermediate 6 or 10. Multiply your final primes back together to check you recover the original number. A repeated prime can be written using a power: 2³ means 2 × 2 × 2.
Why the HCF uses shared building blocks
Suppose 60 red counters and 84 blue counters must be split into the greatest possible number of identical bags with none left over. The number of bags must divide both totals, so you need a common factor. Because you want the greatest number of bags, choose the highest common factor.
60 = 2² × 3 × 5 and 84 = 2² × 3 × 7. Only 2² and 3 are available in both. Their product is 12. Using a factor 5 or 7 would fail for one of the totals. Each of the 12 bags gets 5 red and 7 blue counters. Notice that the HCF gives the number of bags, not the number of counters per bag.
Why the LCM needs enough of every prime
For two signals repeating every 60 and 84 seconds, a shared repeat must be a multiple of both periods. It must therefore contain all the prime factors needed to make either number. Use the highest power of every prime appearing: 2² × 3 × 5 × 7 = 420.
At 420 seconds, the first signal has completed 7 periods and the second 5 periods. A smaller shared repeat cannot omit any required prime factor. The question says they start together; otherwise the first coincidence may need additional reasoning about their start times.
Pause and explain
Why is 18 × 24 not the lowest common multiple of 18 and 24?
Worked example
Two lights flash together at noon. One flashes every 18 seconds and the other every 24 seconds. When do they next flash together?
Show the worked solution
- We need the first time that is a positive multiple of both 18 and 24, so use LCM.
- 18 = 2 × 3² and 24 = 2³ × 3. Take 2³ × 3² = 72.
- Check: 72/18 = 4 and 72/24 = 3, both whole numbers. Seventy-two seconds is 1 minute 12 seconds.
Answer 12:01:12 p.m.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Write 90 and 150 as prime products, then find their HCF.
Give me a hint
90 = 9 × 10; 150 = 15 × 10. Keep splitting until every factor is prime.
Compare my reasoning
- 90 = 2 × 3² × 5; 150 = 2 × 3 × 5².
- The common factors with the smaller powers are 2, 3 and 5.
- HCF = 2 × 3 × 5 = 30.
30
Look for these in your work
- Included only primes in the factorisations.
- Checked that 30 divides both totals.
A teacher has 48 pencils and 72 erasers. Make the greatest number of identical packs using everything. How many packs, and what goes in each?
Give me a hint
The number of packs must divide both stock totals.
Compare my reasoning
- HCF(48,72) = 24.
- 48 ÷ 24 = 2 pencils; 72 ÷ 24 = 3 erasers.
- Check: 24 × 2 = 48 and 24 × 3 = 72.
24 packs, each with 2 pencils and 3 erasers.
Look for these in your work
- Explained why HCF applies.
- Answered both the number of packs and their contents.
A rectangular floor is 168 cm by 120 cm. It must be tiled with identical square tiles, with no cutting or gaps. Find the largest possible side length and the number of tiles.
Give me a hint
The tile side must fit a whole number of times along both dimensions.
Compare my reasoning
- 168 = 2³ × 3 × 7; 120 = 2³ × 3 × 5.
- The largest possible side is HCF = 2³ × 3 = 24 cm.
- There are 168/24 = 7 tiles along one side and 120/24 = 5 along the other: 7 × 5 = 35.
Side 24 cm; 35 tiles.
Look for these in your work
- Translated a geometry context into common factors.
- Multiplied rows by columns rather than adding them.
Common mistakes
- Using HCF just because the question says 'greatest': identify what must divide what.
- Multiplying both numbers always gives a common multiple, but not always the lowest one.
- Counting an intermediate branch as well as the prime leaves.
What can you explain now?
Find the HCF of 48 and 72.
Compare with the explanation
24
48 = 2⁴ × 3 and 72 = 2³ × 3². The shared prime product is 2³ × 3 = 24.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Later, invent one situation that needs HCF and one that needs LCM. Explain the difference without using the words highest or lowest.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources