Overview
What you’ll learn
- Express a whole number as a product of its prime factors, and use that factorisation to find the HCF and LCM of two numbers.
- Decide whether a number is a perfect square or perfect cube from the powers in its prime factorisation.
- Compare and order integers, rational numbers and real numbers using the number line and the symbols <, >, ≤, ≥.
- Apply the laws of indices with positive, negative, zero and fractional indices.
- Convert between ordinary numbers and standard form A × 10ⁿ and calculate with standard form.
- Round numbers to a required number of decimal places or significant figures, and estimate computations before checking with a calculator.
In 30 seconds
- 1A prime number has exactly two factors, 1 and itself; every whole number above 1 factors uniquely into primes.
- 2Write a number as a product of prime powers first: 360 = 2³ × 3² × 5 answers every HCF, LCM, square and cube question about it at once.
- 3HCF takes the lowest power of each shared prime; LCM takes the highest power of every prime that appears.
- 4A number is a perfect square only when every prime index is even, and a perfect cube only when every prime index is a multiple of 3.
- 5Negative indices are reciprocals: a⁻ⁿ = 1/aⁿ. Fractional indices are roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ.
- 6Standard form is A × 10ⁿ with 1 ≤ A < 10; multiplying multiplies the A-parts and adds the indices.
- 7Rounding to significant figures counts from the first non-zero digit, not from the decimal point.
- 8Estimate by rounding each number to 1 significant figure first, then judge whether the exact calculator answer is sensible.
Notes · 1 of 5
Integers, rational and real numbers
The number line holds every number this course uses; the inequality symbols describe where one number sits on it.
The integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. A rational number can be written as a fraction of two integers, so 0.375, −4 and 2⅗ are all rational — their decimals terminate or repeat. The real numbers are every point on the number line, rational and irrational together. On the line, numbers grow from left to right, so −7 < −3: further left means smaller, even though 7 looks bigger than 3.
| Symbol | Reads as | Example | Number-line picture |
|---|---|---|---|
| < | is less than | −3 < −7 is false; −7 < −3 is true | open circle at −3, arrow left |
| > | is greater than | −3 > −7 | open circle at −3, arrow right |
| ≤ | is less than or equal to | x ≤ 4 allows x = 4 | filled circle at 4, arrow left |
| ≥ | is greater than or equal to | x ≥ −1 allows x = −1 | filled circle at −1, arrow right |
Notes · 2 of 5
Prime factorisation, HCF and LCM
One factorisation answers four different questions: HCF, LCM, perfect squares and perfect cubes.
A prime number has exactly two factors. The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. To factorise, divide repeatedly by primes in order (2, then 3, then 5, …) or split with a factor tree, and finish with the index notation: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
- 1Factorise each number fully
84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
- 2HCF: lowest powers of shared primes
Shared primes 2, 3, 7; lowest powers give 2 × 3 × 7 = 42.
- 3LCM: highest powers of all primes
Highest powers give 2² × 3² × 7 = 252.
- 4Sanity check
84 × 126 = 10 584, and HCF × LCM = 42 × 252 = 10 584 — for two numbers, HCF × LCM always equals their product.
| Question type | Wants | Why |
|---|---|---|
| Split into equal groups / largest tile that measures both | HCF | The answer must divide both numbers |
| Events coincide / cycles repeat together | LCM | The answer must be a multiple of both numbers |
| Squares/cubes: smallest multiplier | Missing prime powers | Fill each prime index up to an even value (square) or multiple of 3 (cube) |
Notes · 3 of 5
Indices: the laws and the new powers
Multiply adds indices, divide subtracts, and negative or fractional indices mean reciprocal and root.
| Law | Statement | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 7³ × 7⁴ = 7⁷ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Zero index | a⁰ = 1 (a ≠ 0) | 9⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional index | a^(1/n) = ⁿ√a | 8^(1/3) = 2 |
| General fraction | a^(m/n) = (ⁿ√a)ᵐ | 16^(3/4) = (⁴√16)³ = 8 |
Solving index equations becomes easy once both sides share a base. To solve 2ⁿ = 1/32, write 1/32 = 2⁻⁵, so n = −5. To solve 3^(2x+1) = 81^(x−1), write 81 = 3⁴ so the right side is 3^(4(x−1)); equal powers of 3 force 2x + 1 = 4x − 4, giving x = 5/2. Check: 3⁶ = 729 and 81^(3/2) = 729 ✓.
Notes · 4 of 5
Standard form
A × 10ⁿ with 1 ≤ A < 10 keeps very large and very small numbers comparable and calculable.
A number is in standard form when it is written A × 10ⁿ where 1 ≤ A < 10 and n is an integer. The A-part carries the significant figures; the index carries the size. So 0.000408 = 4.08 × 10⁻⁴ (the decimal point moves 4 places right) and 69 400 000 = 6.94 × 10⁷ (7 places left).
- 1Place the decimal point
After the first non-zero digit to make A between 1 and 10.
- 2Count the jumps
Each place the point moves is one power of 10: left moves give positive n, right moves give negative n.
- 3Check
Convert back mentally: 4.08 × 10⁻⁴ = 0.000408 ✓.
Standard form shines in real contexts: the Earth’s mass is about 5.97 × 10²⁴ kg and the Moon’s about 7.35 × 10²² kg, and one division — 5.97 × 10²⁴ ÷ 7.35 × 10²² ≈ 81.2 — says the Earth is about 81 times heavier. Questions ask for such comparisons "giving your answer in standard form", which is a marking point, not decoration.
Notes · 5 of 5
Rounding, significant figures and estimation
Round to the accuracy asked for, quote answers to 3 significant figures by default, and estimate to sanity-check everything.
Decimal places count digits after the point; significant figures count from the first non-zero digit. The number 0.004809 has three leading zeros, but its first significant figure is the 4. To round to 2 significant figures, keep the 4 and the 8 and look at the next digit: significant figures run 4, 8, 0, 9, and the digit after the 8 is a 0, which is less than 5, so the answer is 0.0048. The same digit rule rounds 0.03749 up to 0.0375 at 3 significant figures, because the digit after the 4 is a 9.
| Rounded to | 47.963 | 0.004809 |
|---|---|---|
| 1 significant figure | 50 | 0.005 |
| 2 significant figures | 48 | 0.0048 |
| 3 significant figures | 48.0 | 0.00481 |
| 2 decimal places | 47.96 | 0.00 |
Estimation rounds every number to 1 significant figure and computes the easy version. To estimate (8.049 × 19.6²) ÷ 0.481, use (8 × 20²) ÷ 0.5 = 3200 ÷ 0.5 = 6400; the exact value is 6428 (3 s.f.), so the estimate is within half a percent. Estimation is not a guess — it is a deliberate coarse calculation, and exam answers must show the rounded numbers you used.
Revise
Key terms
The words examiners expect you to use precisely.
- Prime number
- A whole number above 1 with exactly two factors, 1 and itself.2 is the only even prime.
- Prime factorisation
- Writing a number as a product of primes, using index notation for repeats.360 = 2³ × 3² × 5.
- Highest common factor (HCF)
- The largest number that divides two given numbers; the lowest power of each shared prime.
- Lowest common multiple (LCM)
- The smallest number that both given numbers divide; the highest power of every prime that appears.
- Perfect square
- A number whose prime factorisation has every index even, so it equals an integer squared.
- Rational number
- A number expressible as a fraction of two integers; its decimal terminates or repeats.
Show 6 more key termsShow fewer key terms
- Index (power)
- The number of times a base is multiplied by itself; aⁿ means n copies of a.
- Negative index
- A power that means the reciprocal: a⁻ⁿ = 1/aⁿ.
- Fractional index
- A power whose denominator is a root: a^(m/n) = (ⁿ√a)ᵐ.
- Standard form
- The notation A × 10ⁿ with 1 ≤ A < 10 and n an integer.
- Significant figures
- Digits counted from the first non-zero digit; non-exact answers default to 3 of them.
- Estimation
- A deliberately coarse calculation using 1-significant-figure roundings to predict or check a value.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Treating 1 as a prime number.
1 has only one factor, so it is neither prime nor composite.
Instead: Primes have exactly two factors; factorisations start from 2.
Mistake: Swapping the HCF and LCM rules.
Taking highest powers of shared primes for the HCF of 84 and 126 gives 2² × 3² × 7 = 252, which does not divide 84.
Instead: HCF = lowest powers of shared primes (a divisor); LCM = highest powers of all primes (a multiple).
Show 12 more mistakesShow fewer mistakes
Mistake: Reading −3 as bigger than −7 because 3 is bigger than 7.
On the number line −3 lies to the right of −7, and right means greater.
Instead: Among negatives, the one with the smaller size is the larger number: −3 > −7.
Mistake: Computing −3² as 9.
Without brackets the square attaches to the 3 only: −3² = −(3²) = −9.
Instead: Brackets decide what is squared: (−3)² = 9, −3² = −9.
Mistake: Reading a negative index as a negative value, e.g. 2⁻³ = −8.
The minus in the index means reciprocal, not sign change.
Instead: 2⁻³ = 1/2³ = 1/8, a positive fraction.
Mistake: Adding indices when multiplying different bases, e.g. 2³ × 3² = 6⁵.
The product law applies to a common base only.
Instead: 2³ × 3² = 8 × 9 = 72; unlike bases multiply as they are.
Mistake: Cubing a coefficient without cubing it, e.g. (3x²)³ = 3x⁶.
The power of a product raises every factor: (3x²)³ = 27x⁶.
Instead: Distribute the outer power to number and letter parts alike.
Mistake: Subtracting indices in the wrong direction, e.g. m⁻⁶ ÷ m⁻⁴ = m⁻¹⁰.
Division subtracts the lower index from the upper: m⁻⁶⁻⁽⁻⁴⁾ = m⁻².
Instead: m⁻⁶ ÷ m⁻⁴ = m⁻²; check with a number such as m = 1 (both sides must stay equal).
Mistake: Taking a^(m/n) as ⁿ√(aᵐ) and stopping, or as (aᵐ)ⁿ.
The denominator of the fraction is the root; the numerator is the power.
Instead: a^(m/n) = (ⁿ√a)ᵐ — root first keeps numbers small: 16^(3/4) = (⁴√16)³ = 8.
Mistake: Writing standard form with A outside 1 ≤ A < 10, e.g. 18.4 × 10² or 0.57 × 10⁶.
Standard form fixes the A-part between 1 and 10.
Instead: Re-normalise: 18.4 × 10² = 1.84 × 10³ and 0.57 × 10⁶ = 5.7 × 10⁵.
Mistake: Multiplying standard-form numbers by multiplying the indices, e.g. (2 × 10³) × (4 × 10⁴) = 8 × 10¹².
Multiplication adds indices; it never multiplies them.
Instead: Multiply the A-parts (2 × 4 = 8) and add the indices (10³⁺⁴ = 10⁷): 8 × 10⁷.
Mistake: Rounding 0.004809 to 2 significant figures as 0.00 or 0.01.
Significant figures start at the first non-zero digit; leading zeros never count.
Instead: Keep the first two significant digits (4 and 8) and look at the next (0): 0.0048.
Mistake: Dropping the trailing zero in 48.0 and writing 48 for an answer to 3 significant figures.
48 shows only two significant figures; the zero carries the accuracy claim.
Instead: 47.963 = 48.0 to 3 s.f.; state the zero whenever it is significant.
Mistake: Estimating a subtraction of close numbers after rounding, e.g. 5.862 − 5.857 ≈ 6 − 6 = 0.
Equal roundings cancel a genuinely non-zero difference.
Instead: For close differences, subtract exactly first and round the result.
Check yourself
Quick quiz
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6 multiple-choice questions and structured questions with answer lines. Chapter 1 (N1 Numbers and their operations).
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Chapter 1 Practice Paper — mark scheme
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Model answers with a mark for each creditworthy point — for marking at home or in class.
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Questions
Frequently asked questions
What does Sec 3 E-Math Chapter 1 cover?
It covers syllabus sub-topic N1 of O-Level 4052 / SEC G3 K310 in full: primes and prime factorisation with HCF and LCM, integers and rational numbers, approximation and estimation, standard form, and the laws of indices including negative and fractional powers.
How do I decide between HCF and LCM in a word problem?
Ask what the answer must do. Splitting things into equal shared groups points to the HCF (it divides both numbers). Waiting for two repeating cycles to coincide points to the LCM (it is a multiple of both). Checking that HCF ≤ both numbers ≤ LCM catches most swaps.
Why does 3 significant figures matter so much in E-Math?
The examination instructions make 3 s.f. the default accuracy for every non-exact answer (1 d.p. for angles in degrees). Answers that are correct but wrongly rounded lose the accuracy mark, so rounding discipline is worth marks in every single paper.
Do I need standard form if my calculator shows everything?
Yes. Questions explicitly ask for answers "in standard form" and award a mark for the A × 10ⁿ format itself, and large-number contexts (masses, populations, data storage) are far easier to compare when the size sits in the index.
When is this chapter usually tested in school?
Numbers and their operations sit at the start of the Sec 3 year, so they typically anchor Weighted Assessment 1 (Term 1) and reappear in the end-of-year examination. Scope and weighting vary by school, so check your school’s assessment outline.
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Chapter 2: Ratio, Rate and Speed