Classify numbers, find common factors and describe sets
Recognise number types, use prime factors to find HCF and LCM, and organise relationships with set notation and Venn diagrams.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which factorisation of 84 uses only prime factors?
What you will learn
- Classify natural numbers, integers, primes, squares, cubes, triangle numbers, rational and irrational numbers
- Find prime factors, HCF and LCM
- Use two-set Core and three-set Extended Venn diagrams
Know what kind of number you are working with
Natural numbers here include 0. Integers include positive and negative whole numbers. A prime number has exactly two positive factors; 1 is not prime. Square, cube and triangular numbers follow their named patterns. A rational number is a ratio of integers; an irrational number cannot be written as such a ratio.
A reciprocal multiplies its non-zero number to make 1: the reciprocal of 5 is 1/5. Prime factorisation writes a positive integer as a product of primes, for example 84 = 2² × 3 × 7.
Prime factors and set diagrams answer different questions
For HCF, multiply only common prime factors using the smaller power; for LCM, use each prime factor with the larger power. A Venn diagram groups elements by membership: intersection means both sets, union means either set or both, and complement means outside the chosen set within the universal set.
Core Venn diagrams contain at most two sets. Extended may use three sets and also expects ∈, ∉, the empty set ∅, and subset notation ⊆. Always state the universal set before interpreting a complement.
Pause and explain
What is the correct way to find the highest common factor from prime factorisations?
Worked example
Let U = {1, 2, …, 12}, A be the even numbers and B be the multiples of 3. Find A ∩ B, A ∪ B and A′.
Show the worked solution
- A = {2, 4, 6, 8, 10, 12}; B = {3, 6, 9, 12}.
- The numbers in both sets are 6 and 12, so A ∩ B = {6, 12}.
- Combine the sets without repeating elements for the union, and list the elements of U outside A for the complement.
Answer A ∩ B = {6, 12}; A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12}; A′ = {1, 3, 5, 7, 9, 11}.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →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 84 and 126 as products of primes, then find their HCF and LCM.
Give me a hint
Use the lower common powers for the HCF and the higher powers for the LCM.
Compare my reasoning
- 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
- HCF = 2 × 3 × 7 = 42.
- LCM = 2² × 3² × 7 = 252.
HCF 42; LCM 252.
Look for these in your work
- Both prime products correct
- HCF uses common minimum powers
- LCM uses maximum powers
In U = {1, …, 15}, A is the even numbers, B the multiples of 3 and C the prime numbers. List the pairwise intersections and the elements outside A ∪ B ∪ C.
Give me a hint
Place a number in every set it belongs to; check whether any number belongs to all three.
Compare my reasoning
- A ∩ B = {6, 12}; neither is prime, so the triple intersection is empty.
- A ∩ C = {2}; B ∩ C = {3}.
- Every number from 2 to 15 is even, a multiple of 3 or prime, so only 1 lies outside the union.
A ∩ B = {6, 12}; A ∩ C = {2}; B ∩ C = {3}; outside the union: {1}.
Look for these in your work
- Three-set Extended limit respected
- Intersections checked independently
- Universal set used for the outside region
Two warning lights flash every 84 s and 126 s. They flash together now. When do they next flash together?
Give me a hint
A repeated simultaneous event uses the least common multiple.
Compare my reasoning
- Prime factorise: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
- LCM = 2² × 3² × 7 = 252 seconds.
- 252 seconds is 4 minutes 12 seconds.
They next flash together after 252 s, or 4 min 12 s.
Look for these in your work
- Repeated-event model identified
- LCM calculated
- Time converted
Common mistakes
- Calling 1 a prime number
- Using the smaller powers for an LCM
- Forgetting that a complement depends on U
- Drawing more than two Venn sets for Core
What can you explain now?
Which two numbers belong to both the even numbers and multiples of 3 in U = {1, …, 12}?
Compare with the explanation
6 and 12
Both numbers are divisible by 2 and by 3.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do HCF and LCM use the powers in prime factorisations, and how does a complement depend on U?
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.