Improve Tuition · Year 7 Foundation Maths
Factors and Prime Factors — Lesson & Practice Paper
Year 7 Foundation · 94 marks · 30-minute paper

Factors and Prime Factors

A diagnostic ladder from Year 4 to Higher GCSE.

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

We start with simple factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. work. Then we build up to factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12., prime numbersA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, factor treesA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime., HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12., and harder questions like the ones you’ll meet at GCSE later on.

How to use the booklet

There are five levels. Start at Level 1 and work through in order.

If you get stuck, that’s OK — stop there. The level 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:

The ladder:
· Level 1 / Section A — Year 4–5 (KS2). What a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. is. 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..
· Level 2 / Section B — Year 5–6 (KS2). Prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. trees. Index formShorthand for repeated multiplication. 2³ means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6)..
· Level 3 / Section C — Year 7 (KS3). HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..
· Level 4 / Section D — Year 8–9 / Foundation GCSE (Grade 4–5). Word problems, number of factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4., the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.×LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. shortcut.
· Level 5 / Section E — Year 9 Higher / Higher GCSE (Grade 6–7). Perfect squaresA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power. and cubes from prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).. The shortcut used in reverse.
LEVEL 1 · YEAR 4–5 (KS2) · LESSON 1

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 that is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …). Same as a perfect square. are special. A square numberA number that is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …). Same as a perfect square. 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 that is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36, … (= 1², 2², 3², 4², …). Same as a perfect square., 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 factorHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) is the biggest one.

Worked example
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. 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 HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(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. ✓
LEVEL 2 · YEAR 5–6 (KS2) · LESSON 2

Prime factors, factor trees, and index form

Aim: Write a whole numberA number with no fractional part: 0, 1, 2, 3, 4, … as a productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of its prime factorsA factor that is itself a prime number. The prime factors of 12 are 2 and 3.. (We’ll then write it in index formShorthand for repeated multiplication. 2³ means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6). using powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. like 2³.)

A prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. of a number is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. that is also a prime numberA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, ….

Every whole numberA number with no fractional part: 0, 1, 2, 3, 4, … bigger than 1 can be written as prime factorsA factor that is itself a prime number. The prime factors of 12 are 2 and 3. multiplied together. This is called the number’s prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..

Examples:
· 12 = 2 × 2 × 3
· 30 = 2 × 3 × 5
· 100 = 2 × 2 × 5 × 5
Notice that each factorisation contains only prime numbersA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, multiplied together.

The factor treeA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime. method:

  1. Start with the number at the top.
  2. Split it into any factor pairTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. (two numbers that multiply to give it).
  3. If a branch ends in a primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, circle it and stop that branch.
  4. If a branch ends in a compositeA number with more than 2 factors. 12 is composite (factors 1, 2, 3, 4, 6, 12). The opposite of prime., split it again.
  5. When every branch ends in a primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, multiply the leaves together — that’s the prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..
Factor tree for 60 60 2 30 2 15 3 5 prime (stop here) composite (split again) 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Worked example
Find the prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). of 60.
  1. 60 = 2 × 30.
  2. 2 is primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …. Stop that branch.
  3. Split 30: 30 = 2 × 15.
  4. 2 is primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …. Split 15.
  5. 15 = 3 × 5. Both primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … — stop.
  6. Collect leaves: 2, 2, 3, 5.
  7. So 60 = 2 × 2 × 3 × 5.

Index formShorthand for repeated multiplication. 2³ means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6). is a shorthand for repeated multiplication. A powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. is the small raised number you see in something like 23. It tells you how many copies of the base number to multiply together. The powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. is NOT a normal multiplier.

Examples of powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.:
· 23 means 2 × 2 × 2 = 8.  Read as “two cubed” or “two to the powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of three”.
· 52 means 5 × 5 = 25.  Read as “five squared”.
· 24 means 2 × 2 × 2 × 2 = 16.  Read as “two to the powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of four”.
Watch out — the most common index-form error

23 is NOT 2 × 3 = 6.

23 IS 2 × 2 × 2 = 8.

The raised number tells you HOW MANY copies of the base to multiply together, not what to multiply the base BY.

Worked example
Write 60 in index formShorthand for repeated multiplication. 2³ means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6)..
  1. From the factor treeA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime.: 60 = 2 × 2 × 3 × 5.
  2. 2 appears twice — write 22.
  3. 3 appears once — just 3 (no powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. needed).
  4. 5 appears once — just 5.
  5. So 60 = 22 × 3 × 5.

Try one

Find the prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). of 48 in index formShorthand for repeated multiplication. 2³ means 2 × 2 × 2 = 8 (NOT 2 × 3 = 6)..

Show answer
Divide by the smallest prime each time: 48 = 2×24 = 2×2×12 = 2×2×2×6 = 2×2×2×2×3.
Collect them: four 2s and one 3.
Index form: 2⁴ × 3.
Check: 16 × 3 = 48. ✓
LEVEL 3 · YEAR 7 (KS3) · LESSON 3

HCF and LCM using prime factorisation

Aim: Find HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of any two numbers using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..

In Level 1 you found HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. by listing all the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of both numbers. That works well for small numbers like 12 and 18. But for bigger numbers — like 360 and 168 — the lists get long, and it’s easy to miss one. The faster method uses prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..

Part 1 · The method
A QUICK BRIDGE FROM LEVEL 1 — WHY THIS WORKS

You already know HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6 from the lists method. Let’s see why prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). gives the same answer.

Write each number as a productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …:

  • 12 = 2 × 2 × 3
  • 18 = 2 × 3 × 3

For HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.: a common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. must be built only from primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … that BOTH numbers have. Both have a 2 and both have a 3 — but 12 only has one 3, and 18 only has one 2. So the biggest factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. they can share is 2 × 3 = 6. Same answer as the lists method.

For LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.: a common multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. must contain enough primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … to cover BOTH numbers. To be a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 12 it needs two 2s and one 3. To be a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 18 it needs one 2 and two 3s. Combined: two 2s and two 3s. So LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(12, 18) = 2² × 3² = 36.

The rules below are just shortcuts for what we just worked out by hand.

The two rules:
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of the SHARED primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, each at its SMALLEST powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together..
LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of ALL primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … that appear, each at its LARGEST powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together..
Worked example
Find HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(60, 84) and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(60, 84).
  1. Prime factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5. each number first — use the factor treeA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime. method from Level 2. Here are the trees for 60 and 84:
Factor tree for 60 60 2 30 2 15 3 5

60 = 2 × 2 × 3 × 5
= 22 × 31 × 51

Factor tree for 84 84 2 42 2 21 3 7

84 = 2 × 2 × 3 × 7
= 22 × 31 × 71

primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … (stop here)  ·  compositeA number with more than 2 factors. 12 is composite (factors 1, 2, 3, 4, 6, 12). The opposite of prime. (split again)

  1. Now compare the powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.. Make a table with every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … that appears in either number, and write its powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. in each:
How to read the next two columns:
  • Smallest (for HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.): for each primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, pick the smaller of the two powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. (the lower exponent). If only one number has that primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, write ‘not shared’ — HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. skips it.
  • Largest (for LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.): for each primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, pick the larger of the two powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.. Include every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, even ones in only one number.

Special case: if both numbers have the same powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. for a primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … (like 22 and 22 in our example), then ‘smaller’ and ‘larger’ are both that same value — you just write 22 in both columns.

PrimeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … In 60 In 84 Smallest
(for HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.)		 Largest
(for LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.)
2 22 22 22 22
3 31 31 31 31
5 51 — not shared 51
7 — 71 not shared 71
  1. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.: multiply the ‘Smallest’ column (skip the ‘not shared’ rows).
    HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 22 × 31 = 4 × 3 = 12.
  2. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.: multiply the ‘Largest’ column (include every row).
    LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 22 × 31 × 51 × 71 = 4 × 3 × 5 × 7 = 420.
What if the powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. differ?

In the example above, 60 and 84 happened to share 22 and 31 at the same powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together., so the ‘smallest’ column matched the ‘largest’ column. To see why we actually need the words ‘smallest’ and ‘largest’, look at 12 and 18:

12 = 22 × 31,   18 = 21 × 32.

PrimeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … In 12 In 18 Smallest
(for HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.)		 Largest
(for LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.)
2 22 21 21  (smaller) 22  (larger)
3 31 32 31  (smaller) 32  (larger)

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 21 × 31 = 2 × 3 = 6.

LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 22 × 32 = 4 × 9 = 36.

Now you can see why we say “smallest” and “largest” — when the powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. in the two numbers are different, you actually have to pick.

Prime factors of 60 and 84 60 = 2² × 3 × 5 84 = 2² × 3 × 7 60 84 5 (only in 60) 2 2 3 (shared) 7 (only in 84) HCF = overlap only = 2 × 2 × 3 = 12 LCM = everything in either = 2 × 2 × 3 × 5 × 7 = 420 stripes + solid: 60 dots + dashed: 84

The Venn diagram makes the rule visible: overlap = HCF, everything = LCM. Same as the steps above — just drawn.

Watch out — don’t swap the powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. takes the smallest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. takes the largest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.. Swapping them gives the wrong answer.

Try one

Find HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 18 and 24.

Show answer
Prime factorise: 18 = 2×3², 24 = 2³×3.
HCF = lowest powers shared: 2×3 = 6.
LCM = highest powers: 2³×3² = 72.
Check: 6 × 72 = 432 = 18 × 24. ✓
Part 2 · A useful shortcut

Look at the two numbers we just worked out:

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 12 × 420 = 5040.
60 × 84 = 5040 too.

The same number both ways — and that’s not a coincidence. It works for any two numbers:

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(a, b) = a × b

Why it’s useful. If you know the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., you can find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. by dividing — no second factor treeA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime. needed:

LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = (a × b) ÷ HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.

TRY THE SHORTCUT
You’re told HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(45, 60) = 15. Find LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(45, 60).
  1. 45 × 60 = 2700.
  2. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 2700 ÷ 15 = 180.

One division. No second factor treeA diagram for finding prime factors: split the number into factors, keep splitting until everything is prime..

LEVEL 4 · YEAR 8–9 / FOUNDATION GCSE · LESSON 4

Applying HCF, LCM, and prime factorisation

Aim: Use HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. in word problems. Count the factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of a number from its prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).. Use the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. identity in unfamiliar contexts.

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. or LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.? Look at what the question is asking.

Asks for the BIGGEST equal share, tile, pack, or group? → HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.
(The biggest number that goes into BOTH.)
Example: 24 red sweets + 36 blue sweets → most identical bags = HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(24, 36) = 12.

Asks WHEN two repeating events match up again? → LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.
(The smallest number that BOTH go into.)
Example: Bus A every 12 min, Bus B every 18 min → next together = LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(12, 18) = 36 min.

Worked example (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. context)
A rectangular floor is 168 cm by 120 cm. It is tiled with square tiles of equal size, no gaps, no cutting. What is the largest possible tile size?
  1. The tile side must divide BOTH 168 AND 120 exactly. → HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
  2. 168 = 23 × 3 × 7.
  3. 120 = 23 × 3 × 5.
  4. Shared primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … (smallest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.): 23 and 3.
  5. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 8 × 3 = 24 cm.
  6. Check: 168 = 7 tiles wide. 120 = 5 tiles tall. Total 35 tiles. ✓
Worked example (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. context)
Two trains leave a station together at 8:00 am. One returns every 15 minutes. The other every 25 minutes. When do they next leave together?
  1. They next leave together at the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 15 and 25 minutes after 8:00.
  2. 15 = 3 × 5.
  3. 25 = 52.
  4. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. (all primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, largest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together.) = 3 × 52 = 75 minutes.
  5. 75 minutes after 8:00 is 9:15 am. ✓

Counting factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. from prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).. Once you have n in the form pa × qb × rc, the number of factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of n is (a+1)(b+1)(c+1). Why: each factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. is built by choosing how many copies of each primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … to include (0 to a copies of p, 0 to b copies of q, etc.).

Worked example
How many factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. does 360 have?
  1. Prime factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5.: 360 = 23 × 32 × 5.
  2. Number of factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. = (3+1)(2+1)(1+1).
  3. = 4 × 3 × 2 = 24 factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.. ✓
The shortcut in action.

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(a, b) = a × b.

Use it when one of HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12., a, or b is unknown but the others are given.

Example: if a = 8, HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 4, LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 24, then b = (4 × 24) ÷ 8 = 12.

Try one

Two numbers have productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. 240 and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. 4. What is their LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.?

Show answer
Use HCF × LCM = a × b.
4 × LCM = 240.
LCM = 240 ÷ 4 = 60. ✓
LEVEL 5 · YEAR 9 HIGHER / HIGHER GCSE (GRADE 6–7) · LESSON 5

Perfect squares and cubes from prime factorisation

Aim: Decide whether a number is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power. or cube using its prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).. Find the smallest multiplier that turns a number into a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power. or cube.

A perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power. is an integerA whole number (no fractional or decimal part): 0, 1, 2, 3, … and also negatives −1, −2, −3, … A 'positive integer' means an integer greater than zero. you get by squaring an integerA whole number (no fractional or decimal part): 0, 1, 2, 3, … and also negatives −1, −2, −3, … A 'positive integer' means an integer greater than zero.: 1, 4, 9, 16, 25, 36, 49, … A perfect cubeA whole number you get by cubing an integer. 27 is a perfect cube because 27 = 3³. Prime factorisation has every prime to a power that is a MULTIPLE OF 3. is one you get by cubing: 1, 8, 27, 64, 125, 216, …

The criterion (from prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).):
A number is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power. if and only if every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … in its factorisation has an EVEN powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together..
A number is a perfect cubeA whole number you get by cubing an integer. 27 is a perfect cube because 27 = 3³. Prime factorisation has every prime to a power that is a MULTIPLE OF 3. if and only if every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. is a MULTIPLEA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. OF 3.
WHY THIS WORKS

Suppose n = pa × qb. Then √n = pa/2 × qb/2.

For √n to be a whole numberA number with no fractional part: 0, 1, 2, 3, 4, …, the halved powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. a/2 and b/2 must be whole numbersA number with no fractional part: 0, 1, 2, 3, 4, … — meaning a and b must be even.

Same logic for cubes: ∛n is whole only when every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. divides cleanly by 3.

Worked example
Is 144 a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power.?
  1. Prime factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5.: 144 = 16 × 9 = 24 × 32.
  2. PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 2 is 4 (even). PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 3 is 2 (even).
  3. Both powersThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. are even — so 144 IS a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power..
  4. Check: 144 = 122. ✓
Worked example
Find the smallest k > 0 such that 18k is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power..
  1. Prime factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5. 18: 18 = 2 × 32.
  2. For 18k to be a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power., every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. must be EVEN.
  3. PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 2 is 1 (odd) — need one more 2 to bump to 22.
  4. PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 3 is 2 (even) — fine, leave alone.
  5. So k must contribute exactly one 2: k = 2.
  6. Check: 18 × 2 = 36 = 62. ✓
Worked example
Find the smallest k > 0 such that 24k is a perfect cubeA whole number you get by cubing an integer. 27 is a perfect cube because 27 = 3³. Prime factorisation has every prime to a power that is a MULTIPLE OF 3..
  1. Prime factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5. 24: 24 = 23 × 3.
  2. For 24k to be a perfect cubeA whole number you get by cubing an integer. 27 is a perfect cube because 27 = 3³. Prime factorisation has every prime to a power that is a MULTIPLE OF 3., every primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. must be a MULTIPLEA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. OF 3.
  3. PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 2 is 3 (already a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 3) — fine.
  4. PowerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 3 is 1 — need to bump to 3. Add two more 3s.
  5. So k = 32 = 9.
  6. Check: 24 × 9 = 216 = 63. ✓

Try one

Find the smallest k > 0 such that 50k is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power..

Show answer
50 = 2 × 5².
A square needs even powers; the power of 2 is odd.
Multiply by one more 2: k = 2.
Check: 50 × 2 = 100 = 10². ✓
LEVEL 5 · YEAR 9 HIGHER / HIGHER GCSE (GRADE 6–7) · LESSON 6

The HCF × LCM shortcut, used in reverse

Aim: If you know any three of {HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12., a, b}, find the fourth.

From Level 3, you know the shortcut (also called the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. identity in textbooks):

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(a, b) = a × b

So far we’ve used it forwards: compute HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12., then check that it works.

At Higher level we use it backwards. If you know three of the four numbers in the rule, you can find the fourth by rearrangement.

Worked example
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(n, 12) = 4 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(n, 12) = 60. Find n.
  1. Apply the shortcut: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = a × b.
  2. 4 × 60 = n × 12.
  3. 240 = 12n.
  4. n = 240 ÷ 12 = 20.
  5. Verify: 20 = 22×5 and 12 = 22×3. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 22 = 4 ✓. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 22×3×5 = 60 ✓.
Worked example
Two numbers have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. 6 and productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. 432. Find their LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12..
  1. From the shortcut: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4..
  2. 6 × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 432.
  3. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 432 ÷ 6 = 72. ✓
Watch out — always verify

The algebra always works. But that doesn’t guarantee your n actually gives the stated HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. with the given partner. After you find n, prime-factoriseTo find the prime factorisation. 'Prime factorise 60' means: split 60 into a product of primes, like 60 = 2 × 2 × 3 × 5. it and check both HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. directly.

Higher-grade bonus: the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of any two numbers also divides their sum and their difference. So if HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) = 6, then a + b and a − b are both multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6. This is sometimes the fastest way to rule out wrong answers.

Try one

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) = 5 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(a, b) = 75. If a = 25, find b.

Show answer
HCF × LCM = a × b, so 5 × 75 = 25 × b.
375 = 25b.
b = 375 ÷ 25 = 15.
Check: HCF(25, 15) = 5 and LCM = 75. ✓
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.
Highest common factor (HCF)
The biggest common factor of two numbers. HCF(12, 18) = 6.
Lowest common multiple (LCM)
The smallest number that is a multiple of two numbers. LCM(12, 18) = 36.
Prime factor
A factor that is itself a prime number. The prime factors of 12 are 2 and 3.
Prime factorisation
A number written as a product of its prime factors. 12 = 22 × 3.
Factor tree
A diagram for finding the prime factorisation: split the number into factors, keep splitting until everything is prime.
Index form (or power form)
A shorthand for repeated multiplication. 23 means 2 × 2 × 2 = 8.
Coprime
Two numbers that share no factors other than 1. 5 and 7 are coprime. So are 4 and 9.

Recap before you start the paper

Part Two

Now try the practice questions

45 questions across three sections, 30 minutes when you’re ready. The step-by-step walkthroughs unlock after you mark.

Instructions

Timed practice · 30 minutes

When you’re ready, start the timer

This practice is designed to take 30 minutes. Your tutor will see how long you took. You can work untimed if you prefer.

Section A · Level 1 · Year 4–5 (KS2)
Recognise factors. List the factors of a number. Find factor pairs. Find common factors.
Q1.
Is 4 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 12?
Your answer:
(1 mark)
Q2.
Is 5 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 24?
Your answer:
(1 mark)
Q3.
Is 7 a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 49?
Your answer:
(1 mark)
Q4.
List all six factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 18.
Your answer:
,,,,,
(6 marks)
Q5.
From this list, find all four factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 20: 2, 3, 4, 5, 6, 8, 10.
Your answer:
,,,
(4 marks)
Q6.
What is the largest factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 30 that is smaller than 30 itself?
Your answer: (2 marks)
Q7.
List all five factor pairsTwo numbers that multiply to give n. 3 × 4 = 12, so 3 and 4 are a factor pair of 12. of 36. (For each pair, type just the smaller number — e.g. for pair 4×9 type 4.)
Your answer:
,,,,
(5 marks)
Q8.
36 students go on a field trip. The teachers want to split them into equal groups of more than 1 student but fewer than 36 students. How many different group sizes would work?
Your answer: (2 marks)
Q9.
Find any common factorA number that is a factor of two (or more) given numbers. 6 is a common factor of 12 and 18. of 15 and 20 (greater than 1).
Your answer: (1 mark)
Q10.
Find the highest common factorHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) of 42 and 18.
Your answer: (2 marks)
Section B · Level 2 · Year 5–6 (KS2)
Use factor trees. Write a number as a product of prime factors. Use index form (powers).
Q11.
What does 23 equal?
Your answer: (1 mark)
Q12.
What does 52 equal?
Your answer: (1 mark)
Q13.
What does 34 equal?
Your answer: (2 marks)
Q14.
Write 60 as a productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of four prime factorsA factor that is itself a prime number. The prime factors of 12 are 2 and 3.. Type each prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. in a box (in any order).
Your answer:
,,,
(4 marks)
Q15.
Write 54 as a productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of four prime factorsA factor that is itself a prime number. The prime factors of 12 are 2 and 3.. Type each prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. in a box (in any order).
Your answer:
,,,
(3 marks)
Q16.
81 = 3?. Find the powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together..
Your answer: (3 marks)
Q17.
Find the prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). of 88. How many times does 2 appear?
Your answer: (3 marks)
Q18.
Write 140 as a productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. of four prime factorsA factor that is itself a prime number. The prime factors of 12 are 2 and 3.. Type each prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. in a box (in any order).
Your answer:
,,,
(4 marks)
Q19.
Aisha writes: “23 = 6 because 2 × 3 = 6.” Is she correct?
Your answer:
(2 marks)
Q20.
Find the prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). of 550. How many different primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … appear (not counting repeats)?
Your answer: (3 marks)
Section C · Level 3 · Year 7 (KS3)
Use prime factorisation to find HCF and LCM systematically. The HCF × LCM identity.
Q21.
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 24 and 36.
Your answer: (2 marks)
Q22.
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 60 and 84 using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..
Your answer: (3 marks)
Q23.
Find the smallest number that has BOTH 4 and 18 as factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..
Your answer: (2 marks)
Q24.
Find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 60 and 84 using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..
Your answer: (3 marks)
Q25.
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 90 and 126.
Your answer: (3 marks)
Q26.
HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of two different primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … p and q?
Your answer: (1 mark)
Q27.
Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of three numbers: 24, 36, 60.
Your answer: (3 marks)
Q28.
If a is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of b, what is HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b)?
Your answer:
(2 marks)
Q29.
Find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 18 and 24.
Your answer: (3 marks)
Q30.
For the numbers 12 and 18: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. × LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = ?
Your answer: (3 marks)
Section D · Level 4 · Year 8–9 / Foundation GCSE (Grade 4–5)
Word problems applying HCF and LCM. Number of factors from prime factorisation. Using the identity in context.
Q31.
A rectangular wall is 360 cm by 270 cm. It is tiled with square tiles of the same size, with no gaps and no cutting. What is the largest possible tile size (in cm)?
Your answer: (3 marks)
Q32.
Two buses leave a station together at 9:00 am. Bus A returns every 12 minutes. Bus B returns every 20 minutes. After how many minutes will they next leave together?
Your answer: (3 marks)
Q33.
A baker has 48 muffins and 60 biscuits. They want to pack identical boxes that each contain only muffins or only biscuits (not both), with the same number per box in every box. What is the largest number per box?
Your answer: (2 marks)
Q34.
A number has prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). 23 × 32 × 5. What is the number?
Your answer: (3 marks)
Q35.
A number is written as 22 × 3 × 5. How many different factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. does this number have in total? Hint: if n = pa × qb × rc, the number of factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. is (a+1)(b+1)(c+1).
Your answer: (3 marks)
Q36.
Two numbers have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 4 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 24. If one number is 8, what is the other?
Your answer: (2 marks)
Q37.
A gear has 12 teeth. It meshes with another gear that has 18 teeth. A red dot is painted on a tooth of each gear, lined up at the start. After how many teeth of the 12-tooth gear have passed will the red dots line up again?
Your answer: (3 marks)
Q38.
How many different factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. does 120 have?
Your answer: (3 marks)
Q39.
Find the smallest number divisible byGoes in exactly, with no remainder. 36 is divisible by 4 because 36 ÷ 4 = 9. 12, 15, AND 20.
Your answer: (3 marks)
Q40.
Two numbers have productThe result of multiplying. 5 × 4 = 20, so 20 is the product of 5 and 4. 180 and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. 3. What is their LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.?
Your answer: (3 marks)
Section E · Level 5 · Year 9 Higher / Higher GCSE (Grade 6–7)
Perfect squares and cubes from prime factorisation. The HCF × LCM identity used in reverse to find unknowns. Structural reasoning.
Q41.
Is 196 a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power.?
Your answer:
(2 marks)
Q42.
Is 250 a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power.?
Your answer:
(2 marks)
Q43.
Find the smallest positive integerA whole number (no fractional or decimal part): 0, 1, 2, 3, … and also negatives −1, −2, −3, … A 'positive integer' means an integer greater than zero. k such that 18k is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power..
Your answer: (3 marks)
Q44.
Find the smallest positive integerA whole number (no fractional or decimal part): 0, 1, 2, 3, … and also negatives −1, −2, −3, … A 'positive integer' means an integer greater than zero. k such that 75k is a perfect squareA whole number you get by squaring an integer. 36 is a perfect square because 36 = 6². Prime factorisation has every prime to an EVEN power..
Your answer: (3 marks)
Q45.
Find the smallest positive integerA whole number (no fractional or decimal part): 0, 1, 2, 3, … and also negatives −1, −2, −3, … A 'positive integer' means an integer greater than zero. k such that 72k is a perfect cubeA whole number you get by cubing an integer. 27 is a perfect cube because 27 = 3³. Prime factorisation has every prime to a power that is a MULTIPLE OF 3..
Your answer: (4 marks)
Q46.
Write 720 as 2a × 3b × 5c. Find the value of a + b + c.
Your answer: (3 marks)
Q47.
Two numbers n and 12 have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 4 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 60. Find n.
Your answer: (3 marks)
Q48.
Two numbers have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 8 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 120. One of them is 24. Find the other.
Your answer: (3 marks)
Q49.
How many different factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. does 504 have?
Your answer: (3 marks)
Q50.
Two numbers a and b have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(a, b) = 6. Must a + b be divisible byGoes in exactly, with no remainder. 36 is divisible by 4 because 36 ÷ 4 = 9. 6?
Your answer:
(3 marks)
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