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Factors and Factor Pairs — Lesson & Practice Paper
Foundation entry · Grade 1–4 · 35 questions · 80 marks · take your time

Factors and Factor Pairs

Factors, factor pairs and common factors · Grade 1–4.

This booklet helps you learn about factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., step by step.

What’s a factor?

A factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. is a number that fits exactly into another number — with nothing left over.

Example: 12 ÷ 3 = 4. Three fits exactly into twelve, so 3 is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12.

The factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12 are: 1, 2, 3, 4, 6, 12.

What you’ll learn

By the end of this paper you will be able to recognise factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., list them systematically, find factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12., identify the biggest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18., and apply this to real-life problems. Each section opens with its own learning objective and success criteria — check these before you start the section and again when you finish. HCF using primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … factorisation and LCM are covered in the separate PrimeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … Factorisation and LCM and HCF papers.

How to use the booklet

There are seven short sections (A–G). Sections A–D are calculation questions: recognise factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., list factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12., common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18.. Sections E–G are word problems, easier to harder.

If you get stuck, that’s OK — stop there. The section you stop at tells your tutor exactly where you need help.

TUTOR — MATERIALS TO HAVE READY

For pupils who learn best with physical things to handle:

Sections:
· A. Recognising factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.
· B. Listing factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.
· C. Factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12.
· D. Common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. and biggest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18.
· E. Sharing and grouping word problems
· F. Arranging word problems
· G. Biggest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. word problems
FOUNDATION ENTRY · GRADE 1 · CORE LESSON

What a factor is, and factor pairs

Aim: Find all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of a number. (We’ll also meet factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. and common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18..)

A factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of a number is a whole numberA number with no fractional part: 0, 1, 2, 3, 4, … that divides it exactly. That means there’s nothing left over when you divide.

For example, the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12 are 1, 2, 3, 4, 6, and 12 — each one divides 12 exactly. The factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Key facts about factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.:
  • 1 is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of every number.
  • Every number is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of itself.
  • FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. come in pairs: two numbers that multiply to give it. For 12: 1×12, 2×6, 3×4.

Square numbersA number you get when you multiply a whole number by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …). are special. A square numberA number you get when you multiply a whole number by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …). is what you get by multiplying a number by itself.

Examples: 1, 4, 9, 16, 25, 36, 49, … (= 1×1, 2×2, 3×3, 4×4, …)

For square numbersA number you get when you multiply a whole number by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …)., one factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. has the same number twice. For 25, the factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. is 5×5.

Worked example
List all factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 24.
  1. Try each whole numberA number with no fractional part: 0, 1, 2, 3, 4, … from 1 upwards. If it divides 24 exactly, it’s a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  2. 1 × 24 = 24. So 1 and 24 are factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  3. 2 × 12 = 24. So 2 and 12 are factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  4. 3 × 8 = 24. So 3 and 8 are factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  5. 4 × 6 = 24. So 4 and 6 are factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  6. 5? 24 ÷ 5 = 4.8. Not a whole numberA number with no fractional part: 0, 1, 2, 3, 4, …. So 5 is NOT a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
  7. Stop when the pairs start to repeat. (Next would be 6×4, already found.)
  8. All factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 24: 1, 2, 3, 4, 6, 8, 12, 24.

A common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of two numbers is a number that is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of both. The highest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. (HCF) is the biggest one.

Worked example
Find the HCF of 12 and 18.
  1. FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12: 1, 2, 3, 4, 6, 12.
  2. FactorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 18: 1, 2, 3, 6, 9, 18.
  3. Common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. (in both lists): 1, 2, 3, 6.
  4. Highest is 6. So HCF(12, 18) = 6.

Try one

List all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 20.

Show answer
Find the pairs that multiply to 20: 1×20, 2×10, 4×5.
List them in order: 1, 2, 4, 5, 10, 20.
Check: each one divides 20 exactly. ✓
VOCABULARY
Factor
A whole number that divides another exactly. 3 is a factor of 12 because 12 ÷ 3 = 4.
Factor pair
Two numbers that multiply to give n. 3 × 4 is a factor pair of 12.
Common factor
A number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18.
Biggest common factor
The largest of all the common factors. The biggest common factor of 12 and 18 is 6.
Square number
A number that is a factor pair of itself: 4 × 4 = 16, so 16 is a square number.

Recap before you start the paper

Part Two

Now try the practice questions

35 questions. No time pressure when you’re ready. The step-by-step walkthroughs unlock after you mark.

Instructions

Take your time

No time pressure

Work carefully. Your tutor is looking for understanding, method, and the point where you begin to struggle.

Section A · Recognising factors

Decide whether one number is a factor of another.

LEARNING OBJECTIVE 1 · GRADE 1

Decide whether one number is a factor of another.

Success criteria — I can:
  • use the test “does it divide exactly, with no remainder?”
  • explain that 1 and the number itself are always factors
  • check by dividing and looking for a whole-number answer
Q1.
Is 3 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 15?
Your answer:
(1 mark)
Q2.
Is 4 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 22?
Your answer:
(1 mark)
Q3.
Is 6 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 42?
Your answer:
(1 mark)
Q4.
Is 8 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 56?
Your answer:
(1 mark)
Q5.
Is 9 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 50?
Your answer:
(1 mark)
Section B · Listing factors

List every factor of a number, working systematically.

LEARNING OBJECTIVE 2 · GRADE 1

List all the factors of a number, systematically and without missing any.

Success criteria — I can:
  • find factor pairs starting from 1 and working up
  • list the factors in order from smallest to largest
  • know when to stop — when the two halves of a factor pair meet
Q6.
List all four factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 6.
Your answer:
,,,
(4 marks)
Q7.
List all six factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12.
Your answer:
,,,,,
(6 marks)
Q8.
List all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 15.
Your answer:
,,,
(4 marks)
Q9.
List all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 24.
Your answer:
,,,,,,,
(8 marks)
Q10.
List all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 36.
Your answer:
,,,,,,,,
(9 marks)
Section C · Factor pairs

Identify factor pairs. Find missing factors.

LEARNING OBJECTIVE 3 · GRADE 1

Identify the factor pairs of a number.

Success criteria — I can:
  • complete a factor pair when given one factor (e.g. 3 × ? = 24)
  • count how many factor pairs a number has
  • spot when a number has a square factor pair (n × n)
Q11.
Complete this factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12.: 2 × ? = 18.
Your answer: (1 mark)
Q12.
20 has three factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12.: 1×20, 2×10, 4×5. Type the three larger numbers (the ones in the right-hand side of each pair).
Your answer:
,,
(3 marks)
Q13.
How many factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. does 30 have?
Your answer: (1 mark)
Q14.
16 has a factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. where a number multiplies by itself. What is that number?
Your answer: (1 mark)
Q15.
Find the missing factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.: 7 × ? = 42.
Your answer: (1 mark)
Section D · Common factors

Find common factors. Identify the biggest common factor by listing.

LEARNING OBJECTIVE 4 · GRADE 1–4

Find the common factors of two numbers and the biggest one (HCF) by listing.

Success criteria — I can:
  • list all the factors of each number
  • pick out the numbers that appear in both lists
  • identify the largest of these as the biggest common factor (HCF)
Q16.
List all four common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of 12 and 18.
Your answer:
,,,
(4 marks)
Q17.
Find the common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of 24 and 36 that is bigger than 6.
Your answer: (2 marks)
Q18.
What is the biggest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of 16 and 24?
Your answer: (2 marks)
Q19.
How many common factorsA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. do 30 and 45 have in total?
Your answer: (2 marks)
Q20.
Find the biggest common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of 50 and 75.
Your answer: (2 marks)
Section E · Sharing and grouping

Word problems: share and group equally.

LEARNING OBJECTIVE 5 · GRADE 1

Apply factor knowledge to easy sharing and grouping word problems.

Success criteria — I can:
  • recognise when a real-life problem needs factor reasoning
  • decide if a number of items can be shared equally
  • find the number per share or the number of shares
Q21.
Aisha has 24 sweets to share equally between herself and 3 friends (4 people in total). How many sweets does each person get?
Your answer: (2 marks)
Q22.
A baker has 18 muffins. She arranges them on plates with 3 muffins per plate. How many plates does she fill?
Your answer: (2 marks)
Q23.
A class of 30 children is split into equal groups of 5. How many groups?
Your answer: (1 mark)
Q24.
A florist uses 20 roses, with 4 roses per bouquet. How many bouquets?
Your answer: (1 mark)
Q25.
Daniel has 14 toy cars. Can he share them equally between 3 children with NO leftovers?
Your answer:
(2 marks)
Section F · Arranging in rows and boxes

Word problems: multi-step arrangements.

LEARNING OBJECTIVE 6 · GRADE 2

Apply factor knowledge to multi-step arrangement problems.

Success criteria — I can:
  • decide if items can be arranged in equal rows, boxes or groups
  • find possible row or group sizes by listing factors
  • spot and explain when a number is NOT a factor of the total
Q26.
A class of 36 children is arranged into equal rows of 6 for a photo. How many rows?
Your answer: (1 mark)
Q27.
A rectangular garden has 24 plants arranged in equal rows of 4. How many rows?
Your answer: (1 mark)
Q28.
A baker makes 48 muffins and packs them in boxes of 8. How many boxes?
Your answer: (1 mark)
Q29.
Mia has 15 red and 20 blue beads. She makes identical bracelets, each using some of each colour, with no leftovers. What is the biggest number of bracelets?
Your answer: (2 marks)
Q30.
A teacher has 20 pencils and 30 rubbers to put into identical packs, using ALL of them. What is the biggest number of packs?
Your answer: (2 marks)
Section G · Biggest common factor in context

Word problems: the answer is the biggest common factor.

LEARNING OBJECTIVE 7 · GRADE 4

Find the biggest common factor in real-life contexts (HCF word problems).

Success criteria — I can:
  • recognise a problem asking for “biggest equal share / biggest tile / biggest bag” as HCF
  • find the biggest common factor of the two given numbers by listing
  • check the answer makes sense in the real-life context
Q31.
James has 40 sweets to share equally between his friends so each friend gets MORE than 4 sweets. What is the maximum number of friends?
Your answer: (2 marks)
Q32.
A rectangular floor is 24 cm by 18 cm. It will be tiled with identical square tiles with no cuts. What is the biggest tile size (in cm) that works?
Your answer: (2 marks)
Q33.
A bakery has 28 cakes and 42 buns. They pack identical boxes using ALL of them, with the same number of cakes per box and same number of buns per box. What is the biggest number of boxes?
Your answer: (2 marks)
Q34.
Mr Khan has 96 sweets and 64 chocolates. He makes identical goody bags using ALL of them. What is the biggest number of bags?
Your answer: (2 marks)
Q35.
A school of 60 children arranged in equal rows. There are between 8 and 11 children per row. How many children per row?
Your answer: (2 marks)
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