Learning objectives
Recognise natural numbers, integers, primes, squares, cubes, triangle numbers, rational and irrational numbers and reciprocals
Write a number as a product of prime factors and use it to find the HCF and LCM
Evaluate squares, cubes, roots and other powers with and without a calculator
Use set notation (ℰ, n(A), ∪, ∩, A′, and at Extended ∅, ∈ and ⊆) and count the elements in each region of a Venn diagram
Key definitions
Prime number
A whole number greater than 1 whose only factors are 1 and itself; 2 is the only even prime and 1 is not prime.
Highest common factor (HCF)
The largest number that divides exactly into each of two or more numbers.
Lowest common multiple (LCM)
The smallest number that is a multiple of each of two or more numbers.
Irrational number
A number that cannot be written as a fraction of two integers, so its decimal neither terminates nor recurs, such as √2 and π.
Explanation
The natural numbers are the counting numbers 1, 2, 3, …; the integers add zero and the negatives. A prime has exactly two factors, so the primes start 2, 3, 5, 7, 11, 13; 1 is not prime because it has only one factor. Square numbers (1, 4, 9, 16, 25, …) come from multiplying a number by itself, cube numbers (1, 8, 27, 64, 125, …) from multiplying it by itself twice, and triangle numbers (1, 3, 6, 10, 15, …) from adding 1 + 2 + 3 + …. A rational number can be written as a fraction of two integers, so every terminating or recurring decimal is rational, including 0.75 = 3/4 and 0.333… = 1/3. An irrational number cannot: √2, √10 and π have decimals that never end and never repeat, whereas √49 = 7 is rational. The reciprocal of a number is 1 divided by it: the reciprocal of 4 is 1/4 and the reciprocal of 2/3 is 3/2. Zero has no reciprocal.
Every whole number greater than 1 can be written as a product of prime factors in exactly one way. Use a factor tree, splitting each number until only primes remain: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2³ × 3² × 5. Once two numbers are in this form, the HCF is the product of the primes they share, each taken to the lower power, and the LCM is the product of every prime that appears, each taken to the higher power. For 72 = 2³ × 3² and 120 = 2³ × 3 × 5, the HCF is 2³ × 3 = 24 and the LCM is 2³ × 3² × 5 = 360. In context, the HCF answers "largest equal pieces" questions and the LCM answers "when do they next coincide" questions, such as buses leaving together every 12 and 18 minutes (LCM 36 minutes).
A power tells you how many times to multiply a base by itself: 3⁴ = 3 × 3 × 3 × 3 = 81. The square root reverses squaring, so √81 = 9, and the cube root reverses cubing, so ∛64 = 4 because 4³ = 64. Learn the squares up to 15² = 225 and the cubes 1, 8, 27, 64, 125 and 1000 — the non-calculator papers assume you know them. Remember that squaring a negative number gives a positive result, (−5)² = 25, so x² = 25 has two solutions, x = 5 and x = −5, while the cube root of a negative number is negative: ∛(−27) = −3. On a calculator use the x², xʸ and root keys, and check that the answer is sensible: √50 must lie between √49 = 7 and √64 = 8. Higher roots such as ⁵√32 = 2 are found in the same way, and at Extended any root can be written as a fractional power, which links this work to the rules of indices.
Set notation is a compact language. ℰ is the universal set, the collection of everything under discussion, and n(A) is the number of elements in A. At Extended you also use ∅, the empty set; x ∈ A, meaning x is an element of A; and A ⊆ B, meaning every element of A is also in B. A′ (the complement) contains the members of ℰ that are not in A, A ∪ B (the union) contains anything in A or in B or in both, and A ∩ B (the intersection) contains only what is in both. On a Venn diagram each set is a circle inside a rectangle for ℰ. To fill it in, always start with the overlap, then subtract to find the "only" regions, and finally place the elements in neither set outside the circles. The key counting fact is n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the overlap would otherwise be counted twice. At Extended you meet three overlapping circles: start from the central region and work outwards.
Formulae
HCF = product of the shared prime factors, each to its lower power
LCM = product of all the prime factors that occur, each to its higher power
HCF × LCM = product of the two numbers
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A′) = n(ℰ) − n(A)
Worked examples
q: Find the highest common factor and the lowest common multiple of 84 and 126.
steps: 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7
126 = 2 × 63 = 2 × 3 × 21 = 2 × 3² × 7
HCF: the shared primes at the lower power, 2 × 3 × 7 = 42
LCM: every prime at the higher power, 2² × 3² × 7 = 252 (check: 42 × 252 = 10 584 = 84 × 126)
In context
Bus timetables: two routes leaving a station together every 12 and 18 minutes next leave together after the LCM, 36 minutes.
Tiling and packaging use the HCF to find the largest square tile or box that fits a space exactly with nothing left over.
Common misconceptions
Calling 1 a prime number — it has only one factor, so it is neither prime nor composite.
Thinking every square root is irrational: √36 = 6 and √(1/4) = 1/2 are rational.
Mixing up HCF and LCM — the HCF is never bigger than the smaller number and the LCM is never smaller than the larger number.
Reading A ∪ B as "and": the union means in A or in B (or both); the intersection A ∩ B is the "and".
Study tips
Fill a Venn diagram from the middle outwards: overlap first, then the "only" regions, then the outside.
Check HCF and LCM answers with HCF × LCM = product of the two numbers.
Summary
Primes have exactly two factors; rational numbers are fractions of integers, irrational numbers (√2, π) are not.
Prime factorisation gives the HCF (shared primes, lower powers) and the LCM (all primes, higher powers).
n(A ∪ B) = n(A) + n(B) − n(A ∩ B); A′ is everything in ℰ that is not in A.
Check your understanding
Question 1: Which of these numbers is prime?
1. 51
2. 53
3. 57
Question 2: If n(ℰ) = 40, n(A) = 22, n(B) = 15 and n(A ∩ B) = 7, then n(A ∪ B) is
1. 37
2. 44
3. 30