📖 Theory notes

Every note is built the same way: I can…, a quick recall, 🔑 key ideas, ✨ tips, ⚠️ common mistakes, ✏️ worked examples and 🧠 Remember in 3 lines.

🎯 Learn, then practise from two books

Each LO starts with 📘 a learning sheet (worked examples one step at a time), then 📗 Collins practice (Book 1 exercises) and, where Book 2 covers the topic, 📙 Pearson practice (Book 2 exercises, including starred questions).

🧪 LO tests · 30 minutes each

When the practice feels easy, take the timed test for that LO. Tests run in Quiz mode, submit themselves at 00:00 and can be taken once. Your report shows each LO as a pie chart.

About this chapter

Every whole number is built from prime numbers, the way molecules are built from atoms. This chapter covers multiples, factors, primes, squares and cubes, and then uses prime factors to find the highest common factor (HCF) and lowest common multiple (LCM) — tools you will use for fractions, algebra and real problems such as timetables and packing.

All practice comes from 📗 Collins Book 1, Chapter 1. Pearson Book 2 has no unit on this topic, so this chapter has no 📙 Pearson sheets. The LO tests use Collins "Examination questions on Number" (past Edexcel papers).

LO	📗 Collins Book 1	📙 Pearson Book 2
LO 1	1.1 Multiples · 1.2 Factors (Exercises 1A, 1B, 1E)	—
LO 2	1.3 Primes · 1.4 Squares and cubes (Exercises 1C, 1D, 1E)	—
LO 3	1.5 Products of prime numbers (Exercise 1F)	—
LO 4	1.6 HCF and LCM (Exercise 1G)	—

Learning objectives

LO 1 · 1.1–1.2 Multiples and factorsI can list multiples and factors of a whole number, find factor pairs, and find common multiples and common factors of two numbers.

LO 2 · 1.3–1.4 Primes, squares and cubesI can recognise prime numbers, square numbers and cube numbers, and use their patterns and powers.

LO 3 · 1.5 Products of prime numbersI can write a whole number as a product of prime factors using index notation, and change index form back into a number.

LO 4 · 1.6 HCF and LCMI can find the highest common factor and lowest common multiple of two numbers, using lists or prime factors, and solve problems with them.

Check what you already know

You need	Quick check	Answer
Times tables	7 × 8	56
Division with remainders	45 ÷ 7	6 remainder 3
Odd and even	Is 135 even?	No — it ends in 5
Powers	3 × 3 × 3	27

1.1–1.2 Multiples and factors

LO 1I can… list multiples and factors of a whole number, find factor pairs, and find common multiples and common factors of two numbers.

🔁 Quick recall before you start
What is 6 × 7? What is 42 ÷ 6? (42 and 7 — multiples and factors are two views of the same times table.)

📗 Collins: 1.1 multiples (Exercise 1A), 1.2 factors and factor pairs (Exercise 1B), mixed practice in Exercise 1E.
📙 Pearson Book 2: no matching unit — Book 2 starts after this topic.

🔑 Key idea · Multiples
A multiple of a number is in its times table: multiples of 5 are 5, 10, 15, 20, … The first multiple is the number itself. A common multiple is in both tables: 35, 70, 105, … are common multiples of 5 and 7.

🔑 Key idea · Factors
A factor divides a number exactly. Factors come in pairs: 1 × 20, 2 × 10, 4 × 5, so the factors of 20 are 1, 2, 4, 5, 10, 20. 1 and the number itself are always factors.

✏️ Worked example · Factors of 36 (Collins Example 1)

Pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
6 × 6 repeats 6, so count it once: 1, 2, 3, 4, 6, 9, 12, 18, 36.

✏️ Worked example · Common factors

12: 1, 2, 3, 4, 6, 12 · 20: 1, 2, 4, 5, 10, 20.
Common factors: 1, 2, 4. The highest common factor is 4.

✨ Tip · Stop at the square root
When you search for factor pairs, you can stop once the first number of the pair passes the square root. For 36, stop at 6 — every pair has already appeared.

⚠️ Common mistake · Mixing up the words
Factors are small (they go into the number); multiples are big (the number goes into them). 4 is a factor of 12; 12 is a multiple of 4.

✨ Tip · Divisibility checks
2: ends in an even digit · 3: digit sum divides by 3 · 5: ends in 0 or 5 · 9: digit sum divides by 9 · 10: ends in 0.

🧠 Remember in 3 lines

Multiples: the times table of the number (it never ends).
Factors: numbers that divide in exactly — find them in pairs.
Common = in both lists.

Memory hook: Factors fit in, multiples grow out.

1.3–1.4 Prime, square and cube numbers

LO 2I can… recognise prime numbers, square numbers and cube numbers, and use their patterns and powers.

🔁 Quick recall before you start
How many factors does 7 have? (Two: 1 and 7.) What is 5 × 5? (25.)

📗 Collins: 1.3 prime numbers (Exercise 1C), 1.4 square and cube numbers (Exercise 1D), mixed in Exercise 1E.
📙 Pearson Book 2: no matching unit — Book 2 starts after this topic.

🔑 Key idea · Prime numbers
A prime number has exactly two factors: 1 and itself. Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. 1 is not prime (only one factor). 2 is the only even prime.

🔑 Key idea · Square numbers
A number times itself: 1, 4, 9, 16, 25, 36, … written 1², 2², 3², … They make square patterns of dots.

🔑 Key idea · Cube numbers
A number times itself three times: 1, 8, 27, 64, 125, 216, 1000 = 1³, 2³, 3³, 4³, 5³, 6³, 10³.

n	1	2	3	4	5	6	7	8	9	10
n²	1	4	9	16	25	36	49	64	81	100
n³	1	8	27	64	125	216	343	512	729	1000

✏️ Worked example · Is 1331 a cube number? (Collins Exercise 1D Q6)

Try 11: 11 × 11 = 121, 121 × 11 = 1331.
So 1331 = 11³ — yes.

✏️ Worked example · Patterns from squares

1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16: adding odd numbers gives squares.
2, 5, 10, 17, 26, … is each square + 1.

⚠️ Common mistake · Squared is not doubled
5² means 5 × 5 = 25, not 5 × 2 = 10. 2³ means 2 × 2 × 2 = 8, not 6.

⚠️ Common mistake · Odd does not mean prime
9, 15, 21, 27, 51 and 91 are odd but not prime (91 = 7 × 13). Always test small primes.

✨ Tip · Testing a prime
To test n, try dividing by the primes up to √n. For 97, try 2, 3, 5, 7 — none work, so 97 is prime.

🧠 Remember in 3 lines

Prime: exactly two factors (1 is not prime, 2 is the only even prime).
Square: n × n · Cube: n × n × n.
Know squares to 15² and cubes to 5³ (plus 10³) by heart.

Memory hook: Two factors make a prime; a square has a partner of itself.

1.5 Products of prime numbers

LO 3I can… write a whole number as a product of prime factors using index notation, and change index form back into a number.

🔁 Quick recall before you start
Is 51 prime? (No: 51 = 3 × 17.) What is 2 × 2 × 2 in index form? (2³.)

📗 Collins: 1.5 products of prime numbers (Example 2, Exercise 1F), exam question on 825.
📙 Pearson Book 2: no matching unit — Book 2 starts after this topic.

🔑 Key idea · Every number is a product of primes
Every whole number bigger than 1 can be written as a product of primes in exactly one way: 5472 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 19 = 2⁵ × 3² × 19. Use index notation for repeated primes.

✏️ Worked example · Division ladder (Collins Example 2)

Divide by the smallest prime that works, again and again.
702 ÷ 2 = 351, ÷ 3 = 117, ÷ 3 = 39, ÷ 3 = 13 (prime).
702 = 2 × 3³ × 13.

✏️ Worked example · Factor tree

360 = 36 × 10 = (4 × 9) × (2 × 5) = (2 × 2 × 3 × 3) × (2 × 5).
360 = 2³ × 3² × 5.

✨ Tip · Typing answers
Type powers with ^ and multiply with × or x: 2^3×3^2×5. The order of the primes does not matter.

⚠️ Common mistake · Stopping too early
8 × 9 is not a product of primes. Keep going until every factor is prime: 72 = 2³ × 3².

✨ Tip · Roots from prime factors
A square number has only even powers: 784 = 2⁴ × 7², so √784 = 2² × 7 = 28. Divide the powers by 3 for a cube root.

🧠 Remember in 3 lines

Divide by the smallest prime again and again.
Stop when the last number is prime.
Write repeated primes as powers, smallest prime first.

Memory hook: Break it down until only primes are left.

1.6 HCF and LCM

LO 4I can… find the highest common factor and lowest common multiple of two numbers, using lists or prime factors, and solve problems with them.

🔁 Quick recall before you start
Write 60 as a product of primes. (2² × 3 × 5.) What is the first common multiple of 4 and 6? (12.)

📗 Collins: 1.6 HCF and LCM (Exercise 1G); timetable problems in Exercise 1E; past paper questions 3H Q11 and 4H Q11.
📙 Pearson Book 2: no matching unit — Book 2 starts after this topic.

🔑 Key idea · Highest common factor (HCF)
The biggest number that divides both numbers. With prime factors: take the primes the numbers share, each with the lower power. 60 = 2² × 3 × 5, 72 = 2³ × 3² → HCF = 2² × 3 = 12.

🔑 Key idea · Lowest common multiple (LCM)
The smallest number in both times tables. With prime factors: take every prime, each with the higher power. LCM(60, 72) = 2³ × 3² × 5 = 360.

✏️ Worked example · Together again (Collins Exercise 1E Q4)

Dogs bark every 8 s and every 12 s.
They bark together after LCM(8, 12) = 24 seconds.

✏️ Worked example · Equal groups (HCF)

48 red and 60 blue tiles in identical groups: HCF(48, 60) = 12 groups.
Each group has 4 red and 5 blue tiles.

✏️ Worked example · Finding a missing number (Paper 4H)

LCM(x, 12) = 120 and HCF(x, 12) = 4.
HCF × LCM = x × 12: 480 = 12x, so x = 40.

⚠️ Common mistake · HCF and LCM swapped
The HCF is never bigger than the smaller number; the LCM is never smaller than the bigger number. Check your answer against this.

✨ Tip · Which one do I need?
'Together again', 'at the same time', 'smallest number that both divide into' → LCM. 'Largest', 'equal groups', 'cut into the biggest equal pieces' → HCF.

🧠 Remember in 3 lines

HCF: shared primes, lower powers.
LCM: all primes, higher powers.
HCF × LCM = the two numbers multiplied.

Memory hook: HCF is the overlap; LCM is everything.