Numbers and Their Operations covers the real number system, operations with negative numbers, prime factorisation, HCF and LCM, squares, cubes and roots, and approximation and estimation. Every later topic, including algebra, indices and graphs, is built on knowing exactly what kind of numbers you are working with and how their operations behave.
The whole topic runs on one engine: prime factorisation in index notation. Once a number is factorised, HCF, LCM, perfect squares, perfect cubes and roots are read straight off the indices. This includes the classic “find the smallest value of k” question, which appears in almost every Sec 2 test on this topic.
What these notes cover
- •The real number system
- •Representing and ordering numbers on the number line
- •Sign rules and order of operations
- •Using your calculator correctly
- •Primes and prime factorisation
- •HCF and LCM
- •HCF and LCM word problems
- •Squares, cubes and roots via prime factorisation
- •Rounding to decimal places and significant figures
- •Estimating a calculation
- •Why you must not round too early
- •Common mistakes and exam technique
Key definitions
- Rational number
- A number that can be written as p ÷ q, where p and q are integers and q ≠ 0. Every integer is rational (−7 = −7 ÷ 1), and every terminating or recurring decimal is rational.
- Irrational number
- A real number that cannot be written as p ÷ q with integers p and q. Its decimal is non-terminating and non-recurring, e.g. π = 3.14159… and √2 = 1.41421…. A root sign alone does not make a number irrational: √49 = 7.
- Prime number
- A whole number greater than 1 that has exactly two distinct factors: 1 and the number itself. 1 is not prime, and 2 is the only even prime.
- Highest common factor (HCF)
- The largest number that divides both numbers exactly, e.g. HCF(12, 18) = 6.
- Lowest common multiple (LCM)
- The smallest number that both numbers divide into exactly, e.g. LCM(12, 18) = 36.
Key formulas
Rational number
p and q must be integers. The condition q ≠ 0 matters because division by zero is undefined.
Sign rules
Same signs give a positive answer and different signs give a negative answer. Division follows the same rules.
Prime factorisation in index notation
p < q < r are primes in ascending order, and the indices a, b, c count how many times each prime appears, e.g. 1176 = 2³ × 3 × 7². This is the expected final form.
HCF–LCM relationship
HCF(a, b) × LCM(a, b) = a × b
This holds for two positive integers only and fails for three or more. Use it to check answers, or to find LCM = (a × b) ÷ HCF.
Square root from prime factorisation
Halve every index. A number is a perfect square exactly when every index is even, e.g. √(2⁴ × 3²) = 2² × 3 = 12.
Cube root from prime factorisation
Divide every index by 3. A number is a perfect cube exactly when every index is a multiple of 3, e.g. ∛(2⁶ × 5³) = 2² × 5 = 20.
Worked example
It is given that N = 600 = 2³ × 3 × 5². (i) Find the smallest positive integer k such that Nk is a perfect square. (ii) Find the smallest positive integer m such that Nm is a perfect cube.
- 1Write every index explicitly: N = 2³ × 3¹ × 5².
- 2(i) For a perfect square every index must be even, so the index of 2, which is 3 (odd), needs one more factor of 2 to reach 2⁴.
- 3The index of 3 is 1 (odd), so it needs one more factor of 3 to reach 3², and the index of 5 is 2, which is already even.
- 4So the smallest multiplier is k = 2 × 3 = 6.
- 5Verify: Nk = 600 × 6 = 3600 = 2⁴ × 3² × 5², and halving the indices gives √3600 = 2² × 3 × 5 = 60.
- 6(ii) For a perfect cube every index must be a multiple of 3, and the index of 2 is already 3, so no factor of 2 is needed.
- 7The index of 3 is 1, so it needs 3² more to reach 3³, and the index of 5 is 2, so it needs one more 5 to reach 5³.
- 8So m = 3² × 5 = 45.
- 9Verify: Nm = 600 × 45 = 27000 = 2³ × 3³ × 5³, and dividing each index by 3 gives ∛27000 = 2 × 3 × 5 = 30.
Watch out: An unwritten index is 1, and 1 is odd. Skipping the prime 3 in 2³ × 3 × 5² gives the wrong smallest k.
Common mistakes
Squaring a negative without brackets
Wrong: Evaluating (−3)² + 4 by keying in −3² + 4 and writing down −5.
Right: −3² = −(3²) = −9, but (−3)² = (−3) × (−3) = 9. Key in the brackets: (−3)² + 4 = 9 + 4 = 13.
Swapping the HCF and LCM rules
Wrong: With 72 = 2³ × 3² and 60 = 2² × 3 × 5, writing HCF = 2³ × 3² × 5 = 360.
Right: HCF = 2² × 3 = 12 (common primes, lowest power) and LCM = 2³ × 3² × 5 = 360 (every prime, highest power). The HCF can never be larger than the smaller number.
Rounding intermediate values
Wrong: For a square field of side 250 ÷ 7 m, taking side ≈ 35.7 m, so area = 35.7² = 1274.49 ≈ 1270 m² (3 s.f.).
Right: Keep full calculator accuracy: side = 35.71428571… m, so area = 1275.510… ≈ 1280 m² (3 s.f.). Keep at least 4–5 s.f. in working and round only the final answer.
Questions students ask
Is 1 a prime number?
No. A prime has exactly two distinct factors, 1 and itself, but 1 has only one factor, so it fails the definition. It is not composite either. The smallest prime is 2, which is also the only even prime.
How do you find the HCF and LCM using prime factorisation?
Write each number in index notation first. The HCF takes the lowest power of each common prime, and the LCM takes the highest power of every prime that appears in either number. For example, 360 = 2³ × 3² × 5 and 168 = 2³ × 3 × 7 give HCF = 2³ × 3 = 24 and LCM = 2³ × 3² × 5 × 7 = 2520.
How do I know whether to use HCF or LCM in a word problem?
Use HCF when you are cutting, splitting or sharing something into the largest possible identical pieces or groups, e.g. “largest square tile” or “greatest possible length”. Use LCM when separate repeating events must come together again, or when you need the smallest common quantity, e.g. “next flash together”. HCF answers are smaller than the given numbers, and LCM answers are larger.
Do leading zeros count as significant figures?
No. Leading zeros only fix the position of the decimal point, so 0.00583 has 3 s.f. Zeros between non-zero digits always count (4.007 has 4 s.f.), and so do trailing zeros after a decimal point (2.500 has 4 s.f.).
In the full PDF
- •Factor-tree and repeated-division diagrams for 1176, 936 and 1350, plus the number-line figure
- •Further worked examples on classifying numbers, ordering negatives, evaluating with negatives and fractions, and testing divisibility from a factorisation
- •HCF and LCM word problems on tiling a hall floor, flashing lights, goodie bags and cutting ribbons
- •The remaining pitfalls, including treating 1 as prime, claiming π equals 22 ÷ 7, applying HCF × LCM to three numbers and miscounting significant figures
- •Calculator technique for the fraction key, the (−) key and bracketing negatives
- •Section summaries and the five key takeaways for the exam