Number

What you'll learn

This guide covers all Number topics examined in WJEC GCSE Mathematics, including operations with integers, decimals and fractions, working with percentages, prime factorisation, multiples and factors, powers and roots, and standard form. These foundational skills underpin every other area of mathematics and frequently appear in context-based questions across the entire exam paper.

Key terms and definitions

Prime number — a number greater than 1 that has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11).

HCF (Highest Common Factor) — the largest number that divides exactly into two or more numbers without a remainder.

LCM (Lowest Common Multiple) — the smallest number that is a multiple of two or more numbers.

Standard form — a method of writing very large or very small numbers in the format a × 10^n where 1 ≤ a < 10 and n is an integer.

Recurring decimal — a decimal number in which a digit or group of digits repeats infinitely (e.g. 0.3̇ = 0.333...).

Reciprocal — the multiplicative inverse of a number; for any non-zero number x, the reciprocal is 1/x.

Index — the power to which a number is raised (in 5³, the index is 3).

Product of prime factors — expressing a number as a multiplication of prime numbers only.

Core concepts

Types of numbers and their properties

Integers are whole numbers, including negative numbers, zero and positive numbers. They form the basis of many calculations.

Rational numbers can be expressed as a fraction p/q where p and q are integers and q ≠ 0. All integers, fractions, terminating decimals and recurring decimals are rational.

Irrational numbers cannot be expressed as exact fractions. Examples include π, √2, and √3. These have non-recurring, non-terminating decimal expansions.

Key number sets you must recognise:

• Natural numbers: 1, 2, 3, 4...
• Square numbers: 1, 4, 9, 16, 25...
• Cube numbers: 1, 8, 27, 64...
• Triangle numbers: 1, 3, 6, 10, 15...

Factors, multiples and primes

Finding factors means identifying all numbers that divide exactly into a given number. Factors always come in pairs.

For 24: 1×24, 2×12, 3×8, 4×6 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Finding multiples means listing numbers in the times table of a given number.

Multiples of 7: 7, 14, 21, 28, 35...

Prime factorisation breaks any number into a product of prime factors. Use a factor tree or repeated division:

For 72:

• 72 = 2 × 36
• 36 = 2 × 18
• 18 = 2 × 9
• 9 = 3 × 3

Therefore: 72 = 2³ × 3²

Finding the HCF: Write both numbers as products of primes, then multiply the common prime factors.

For HCF of 72 and 120:

• 72 = 2³ × 3²
• 120 = 2³ × 3 × 5
• HCF = 2³ × 3 = 24

Finding the LCM: Write both numbers as products of primes, then multiply the highest power of each prime that appears.

For LCM of 72 and 120:

• 72 = 2³ × 3²
• 120 = 2³ × 3 × 5
• LCM = 2³ × 3² × 5 = 360

Operations with fractions, decimals and percentages

Fraction calculations require different methods depending on the operation:

Addition and subtraction: Find a common denominator, then add/subtract numerators.

3/4 + 2/5 = 15/20 + 8/20 = 23/20 = 1 3/20

Multiplication: Multiply numerators together and denominators together, then simplify.

2/3 × 5/7 = 10/21

Division: Multiply by the reciprocal of the second fraction.

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8

Converting between forms:

Fraction to decimal: Divide numerator by denominator. 3/8 = 3 ÷ 8 = 0.375

Decimal to percentage: Multiply by 100. 0.375 = 37.5%

Percentage to decimal: Divide by 100. 37.5% = 0.375

Recurring decimals to fractions:

For 0.7̇ (which equals 0.777...):

• Let x = 0.777...
• Then 10x = 7.777...
• Subtract: 10x - x = 7
• Therefore 9x = 7
• So x = 7/9

For 0.2̇7̇ (which equals 0.272727...):

• Let x = 0.272727...
• Then 100x = 27.272727...
• Subtract: 100x - x = 27
• Therefore 99x = 27
• So x = 27/99 = 3/11

Percentage calculations

Finding a percentage of an amount: Convert the percentage to a decimal or fraction, then multiply.

Find 35% of £240: 0.35 × 240 = £84

Percentage increase and decrease: Use a multiplier for efficiency.

Increase £120 by 15%: Multiplier = 1.15 £120 × 1.15 = £138

Decrease £80 by 12%: Multiplier = 0.88 £80 × 0.88 = £70.40

Reverse percentage (finding the original amount): Identify what percentage the given amount represents, then scale appropriately.

After a 20% increase, a phone costs £360. Find the original price. £360 represents 120% 1% = 360 ÷ 120 = 3 100% = 3 × 100 = £300

Percentage change: Use the formula: (change ÷ original) × 100

A jacket's price rises from £45 to £54: Change = £9 Percentage change = (9 ÷ 45) × 100 = 20% increase

Powers, roots and indices

Index laws are essential for simplifying expressions:

• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• a^0 = 1 (where a ≠ 0)
• a^−n = 1/a^n
• a^(1/n) = ⁿ√a
• a^(m/n) = (ⁿ√a)^m

Examples:

• 5² × 5³ = 5⁵ = 3125
• 7⁴ ÷ 7² = 7² = 49
• (2³)² = 2⁶ = 64
• 4^−2 = 1/4² = 1/16
• 27^(1/3) = ³√27 = 3
• 8^(2/3) = (³√8)² = 2² = 4

Estimating powers and roots: Know perfect squares up to 15² = 225 and perfect cubes up to 5³ = 125.

Estimate √50: Since 7² = 49 and 8² = 64, √50 ≈ 7.1

Surds are irrational roots left in exact form (√2, √3, √5, etc.).

Simplify surds by finding square factors: √48 = √(16 × 3) = 4√3

Multiply surds: √2 × √8 = √16 = 4

Rationalise denominators by multiplying top and bottom by the surd: 1/√3 = 1/√3 × √3/√3 = √3/3

Standard form

Standard form (scientific notation) expresses numbers as a × 10^n where 1 ≤ a < 10.

Large numbers have positive powers: 35,000 = 3.5 × 10⁴

Small numbers have negative powers: 0.00042 = 4.2 × 10^−4

Calculations in standard form:

Multiplication: Multiply the front numbers, add the powers. (3 × 10⁵) × (4 × 10²) = 12 × 10⁷ = 1.2 × 10⁸

Division: Divide the front numbers, subtract the powers. (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴

Addition/subtraction: Convert to the same power of 10 first. (5.3 × 10⁴) + (2.1 × 10³) = (5.3 × 10⁴) + (0.21 × 10⁴) = 5.51 × 10⁴

Calculator use: Enter using the ×10^x or EXP button, never as "× 1 0 ^".

Worked examples

Example 1: Product of prime factors and HCF/LCM

Question: a) Express 180 as a product of prime factors in index form. [2 marks] b) Find the HCF of 180 and 252. [2 marks] c) Find the LCM of 180 and 252. [2 marks]

Solution:

a) Factor tree for 180: 180 = 2 × 90 = 2 × 2 × 45 = 2 × 2 × 9 × 5 = 2 × 2 × 3 × 3 × 5 180 = 2² × 3² × 5 ✓✓

b) First find 252 as product of primes: 252 = 2² × 3² × 7 ✓

HCF = product of common prime factors with lowest powers HCF = 2² × 3² = 4 × 9 = 36 ✓

c) LCM = product of all prime factors with highest powers LCM = 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1260 ✓✓

Example 2: Percentage increase with reverse percentage

Question: The population of a town increased by 12% from 2020 to 2023. In 2023, the population was 28,560. Calculate the population in 2020. [3 marks]

Solution:

28,560 represents 112% (original 100% + 12% increase) ✓

To find 100%: 1% = 28,560 ÷ 112 = 255 ✓ 100% = 255 × 100 = 25,500 ✓

Population in 2020 = 25,500

Example 3: Standard form calculation

Question: The mass of Earth is 5.97 × 10²⁴ kg. The mass of Mars is 6.39 × 10²³ kg. How many times heavier is Earth than Mars? Give your answer in standard form to 2 significant figures. [3 marks]

Solution:

(5.97 × 10²⁴) ÷ (6.39 × 10²³) ✓

= (5.97 ÷ 6.39) × 10²⁴⁻²³ = 0.934... × 10¹ ✓ = 9.34... × 10⁰ = 9.3 (to 2 s.f.) or 9.3 × 10⁰ ✓

Earth is approximately 9.3 times heavier than Mars.

Common mistakes and how to avoid them

• Multiplying percentages incorrectly for successive changes: A 10% increase followed by a 10% decrease does NOT return to the original value. Use multipliers: 1.1 × 0.9 = 0.99, representing an overall 1% decrease. Always multiply the multipliers together.

• Forgetting to rationalise the denominator: Answers with √ symbols in the denominator are not in simplest form. Always multiply by the surd over itself: 5/√2 must become 5√2/2.

• Adding/subtracting fractions with different denominators: You cannot add 1/3 + 1/4 to get 2/7. Find a common denominator (12) to get 4/12 + 3/12 = 7/12.

• Misapplying index laws across addition: You cannot simplify 3² + 3³ using index laws. Index laws only apply to multiplication and division. Calculate: 9 + 27 = 36.

• Writing standard form incorrectly: The number 45 × 10³ is NOT in standard form because 45 ≥ 10. Convert to 4.5 × 10⁴. Always ensure the first number is between 1 and 10.

• Confusing HCF and LCM: HCF is always smaller than or equal to both numbers; LCM is always larger than or equal to both numbers. The HCF divides into both; both divide into the LCM.

Exam technique for "Number"

• Command words matter: "Express" means write in a specific form (e.g., product of primes); "Calculate" requires a numerical answer; "Show that" requires every step of working to reach the given answer exactly; "Estimate" means round first, then calculate.

• Show all working clearly: Number questions typically award method marks even if the final answer is wrong. Write each step on a new line. For 3-mark questions, expect to show at least three clear stages of working.

• Use the calculator efficiently: Know the fraction, power and standard form buttons. For complex calculations, use bracket keys to ensure correct order of operations. Check your answer is reasonable—if asked for a percentage increase and you calculate 0.15, you've forgotten to multiply by 100.

• Check your answer makes sense: If calculating 15% of £80 gives £120, you know something is wrong. Does the answer have appropriate units? Is it positive when it should be? Is the magnitude reasonable?