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LCM and HCF — Lesson & Practice Paper
Foundation GCSE · Grade 4–6 · 38 questions · take your time

LCM and HCF

LCM, HCF, prime factorisation method and word problems · Grade 4–6.

This booklet helps you learn about LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., step by step.

What’s HCF?

HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. means Highest Common FactorHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.. It’s the BIGGEST factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. that two numbers both share.

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, 12. The factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of 18 are 1, 2, 3, 6, 9, 18. The biggest number that appears in both lists is 6. So HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6.

What’s LCM?

LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. means Lowest Common MultipleLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.. It’s the SMALLEST multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. that two numbers both share.

Example: multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 4 are 4, 8, 12, 16, 20, … MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6 are 6, 12, 18, 24, … The smallest number in both lists is 12. So LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(4, 6) = 12.

What you’ll learn

We start by finding LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. using lists. Then we use prime factorA factor that is itself a prime number. The prime factors of 12 are 2 and 3. trees (a faster method). Then we use these skills to solve real-life word problems, and finally tackle some harder GCSE-style questions.

How to use the booklet

There are four sections (A–D). Start at Section A and work through in order.

By the end of this paper you will be able to find LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. by listing and by prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)., and apply these to real-life and Higher-tier problems. Each section opens with its own learning objective and success criteria. If you get stuck, that’s OK — stop there. The section you stop at tells your tutor exactly where you need help.

The four sections:
· Section A. Find LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. by listing.
· Section B. Find LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3)..
· Section C. Word problems and applications.
· Section D. Higher reasoning with 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..
TUTOR — MATERIALS TO HAVE READY

Useful objects for this paper:

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

LCM and HCF by listing

Aim: Find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of two numbers by listing their multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12.. Find the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. by listing their factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4..

A multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of n is a number you get by multiplying n by a whole numberA number with no fractional part: 0, 1, 2, 3, 4, …. MultiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6 are 6, 12, 18, 24, 30, …

The Lowest Common MultipleLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.) of two numbers is the smallest number that appears in BOTH of their times tables.

Worked example
Find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 5 and 6.
  1. List multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5: 5, 10, 15, 20, 25, 30, 35, …
  2. List multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6: 6, 12, 18, 24, 30, 36, …
  3. Smallest number in both lists: 30.
  4. So LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(5, 6) = 30. ✓

A factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of n is a whole numberA number with no fractional part: 0, 1, 2, 3, 4, … that divides n exactly. 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.

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 two numbers is the biggest number that divides BOTH of them exactly.

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. List 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. List 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. Largest: 6. So HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6. ✓
Quick checks:
· If two numbers are different primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … (e.g. 5 and 7), HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 1 and 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..
· If one number is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of the other (e.g. 4 and 12), HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = the smaller and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = the larger.
· LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. is always ≥ the larger number. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. is always ≤ the smaller number.

Try one

Find the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 8 and 10.

Show answer
Multiples of 8: 8, 16, 24, 32, 40; of 10: 10, 20, 30, 40.
First shared: LCM = 40.
Factors shared by 8 and 10 are 1 and 2, so HCF = 2. ✓
PRIME FACTORISATIONA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3). METHOD · LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. AND HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.

LCM and HCF via prime factorisation

Aim: Find LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of any two numbers using prime factorisationA number written as a product of its prime factors. 12 = 2 × 2 × 3 (or 2² × 3).. (Especially useful when the numbers are too big to list factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. easily.)

Earlier you found LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. by listing multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. or factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of each number. That works well for small numbers. But for bigger numbers — like HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(168, 360) — 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)..

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..
A QUICK BRIDGE FROM THE LISTING METHOD — WHY THIS WORKS

Earlier you found HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6 by listing factorsA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4.. 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.

  • 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 from primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … that BOTH numbers have. Both share a 2 and a 3 — but 12 only has one 3, and 18 only has one 2. So the biggest shared factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. is 2 × 3 = 6. Same answer as listing.

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 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 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. = 22 × 32 = 36.

Worked example
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 60 and 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.
  2. 60 = 22 × 3 × 5.
  3. 84 = 22 × 3 × 7.
  4. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.: shared primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … are 2 and 3. Smallest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 2 is 22. Smallest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of 3 is 31.
  5. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 22 × 3 = 4 × 3 = 12.
  6. 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, … are 2, 3, 5, 7. Largest powerThe small raised number in something like 2³. Tells you how many copies of the base to multiply together. of each: 22, 3, 5, 7.
  7. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 22 × 3 × 5 × 7 = 4 × 105 = 420.
Useful identity: for any two numbers a and b, 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.
Check: 12 × 420 = 5040 and 60 × 84 = 5040. ✓
Watch out — smallest vs largest

The 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. of each SHARED primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …. The 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. of ALL primesA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …. Swapping these gives wrong answers.

Fall back to the reasoning if you forget: HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. is a factorA whole number that divides another exactly, with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4. of BOTH numbers, so it can’t have more of any primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, … than the smaller-supply number has.

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 30 and 42.

Show answer
Prime factorise: 30 = 2×3×5, 42 = 2×3×7.
HCF = shared primes: 2×3 = 6.
LCM = 2×3×5×7 = 210. ✓
WORD PROBLEMS · LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. AND HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. IN CONTEXT

Word problems and applications

Aim: Recognise when a word problem needs LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. or HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6., and solve it. (We’ll also 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. shortcut and date arithmetic.)

In Levels 1 and 2 you learned to compute LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. and HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.. That’s the easy part — the maths follows a recipe. The harder part in exams is reading a word problem and spotting WHICH one (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. or HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) the question is actually asking for. The trick is to look at what the question is asking.

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 (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. context)
Lars goes swimming every 2 days. Rita goes every 3 days. Alan goes every 5 days. They all swam together on Friday 1st June. On what date and day will they all next swim together?
  1. "All together again" → LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12..
  2. LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(2, 3, 5). All three are primeA whole number bigger than 1 with exactly 2 factors: 1 and itself. 2, 3, 5, 7, 11, …, so LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 2 × 3 × 5 = 30 days.
  3. 30 days after 1st June. June has 30 days, so day 30 from 1st June is 1st July.
  4. Day of week: 30 ÷ 7 = 4 remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. 2. So 4 full weeks (back to Friday) plus 2 days → Sunday.
  5. Answer: Sunday 1st July. ✓
Date arithmetic shortcut: to find the day of the week N days later, compute N mod 7 (the remainderWhat’s left over after dividing. 7 ÷ 2 = 3 with remainder 1. when dividing by 7). That tells you how many days past the original day of the week to count.
Examples: 14 days later = same day. 15 days later = +1 day. 30 days later = +2 days (since 30 = 4×7 + 2).
Worked example (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. context)
A floor measures 360 cm by 240 cm. It is tiled with square tiles of equal size, with no gaps and no cutting. What is the largest tile size possible?
  1. Tile must divide BOTH dimensions exactly. → HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
  2. 360 = 23 × 32 × 5.
  3. 240 = 24 × 3 × 5.
  4. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 23 × 3 × 5 = 120 cm.
  5. Check: 360 = 3 tiles wide, 240 = 2 tiles tall. Total 6 tiles. ✓
Worked example (identity in reverse)
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. 60. One is 20. Find the other.
  1. Identity: 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 = 20 × b.
  3. 240 = 20b. b = 12. ✓

Try one

Two cogwheels with 12 and 18 teeth start lined up. After how many teeth pass the 12-tooth cog before they line up again?

Show answer
They re-align after a common multiple of 12 and 18.
LCM(12, 18) = 36.
So 36 teeth pass (3 turns of the 12-tooth cog). ✓
HIGHER REASONING · 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. AS TOOLS

HCF and LCM in other topics

Aim: Recognise when 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. is the tool to use inside a problem from a different topic (fractions, ratios, geometry, algebra).

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. are not just standalone topics — they appear inside many other questions as a tool. Spotting which one to use, and applying it correctly, is a Higher GCSE skill.

Where they appear:
· Fractions — simplifying: divide top and bottom by their HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. to get lowest terms.
· Fractions — adding/subtracting: the common denominator is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of the two denominators.
· Ratios — simplifying: divide both parts by their HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6..
· Geometry — constraints: “largest tile that fits” needs HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.; “smallest length that’s a multipleA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of two given lengths” needs LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12..
· Algebra/number theory — finding pairs: if you know 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 two numbers, the identity 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 plus coprimeTwo numbers that share no factors other than 1. 5 and 7 are coprime. So are 4 and 9. reasoning often pins down the pair.
Worked example (FRACTIONS — SIMPLIFYING)
Simplify 12/18 to lowest terms.
  1. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(12, 18) = 6.
  2. 12 ÷ 6 = 2.
  3. 18 ÷ 6 = 3.
  4. So 12/18 = 2/3 in lowest terms. ✓
Worked example (FRACTIONS — ADDING)
Add 1/4 + 1/6.
  1. Common denominator: LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(4, 6) = 12.
  2. 1/4 = 3/12 (multiply top and bottom by 3).
  3. 1/6 = 2/12 (multiply top and bottom by 2).
  4. Add: 3/12 + 2/12 = 5/12. ✓
Worked example (RATIOS)
Simplify the ratio 24 : 36.
  1. HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.(24, 36) = 12.
  2. 24 ÷ 12 = 2.
  3. 36 ÷ 12 = 3.
  4. So 24 : 36 = 2 : 3. ✓
THE BIG IDEA

In every case above, the 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. is doing the same job it did in the word-problem section — finding the biggest equal share (HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6.) or the smallest common quantity (LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.). The challenge is recognising that the question is asking for it, even when the words “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.” never appear in the problem.

Try one

Simplify the fraction 30/45.

Show answer
HCF(30, 45) = 15.
Divide both by 15: 30÷15 = 2, 45÷15 = 3.
30/45 = 2/3. ✓
VOCABULARY
LCM (Lowest Common Multiple)
The smallest number that is a multiple of two (or more) given numbers. LCM(4, 6) = 12.
HCF (Highest Common Factor)
The biggest number that divides two (or more) given numbers exactly. HCF(12, 18) = 6.
Multiple
A number in a times table. 18 is a multiple of 6 because 6 × 3 = 18.
Factor
A whole number that divides another exactly. 3 is a factor of 12.
Prime factorisation
A number written as a product of primes. 12 = 22 × 3.
Coprime
Two numbers that share no factors other than 1. HCF = 1.

Recap before you start

Part Two

Now try the practice questions

38 questions across four sections. No time pressure. 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 · LCM and HCF by listing

Find the lowest common multiple and highest common factor by listing.

LEARNING OBJECTIVE 1 · GRADE 4

Find LCM and HCF of two numbers by listing.

Success criteria — I can:
  • list multiples of each number in order
  • list factors of each number in order
  • pick the smallest number that appears in both multiple lists (LCM)
  • pick the largest number that appears in both factor lists (HCF)
Q1.
List the first five multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 6, starting at 6.
Your answer:
,,,,
(5 marks)
Q2.
List the first five multiplesA number you get by multiplying. 12 is a multiple of 3 because 3 × 4 = 12. of 5, starting at 5.
Your answer:
,,,,
(5 marks)
Q3.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 5 and 6?
Your answer: (2 marks)
Q4.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 3 and 5?
Your answer: (2 marks)
Q5.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 6 and 8?
Your answer: (2 marks)
Q6.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 3 and 5?
Your answer: (1 mark)
Q7.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 6 and 8?
Your answer: (2 marks)
Q8.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 10 and 15?
Your answer: (2 marks)
Q9.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 10 and 15?
Your answer: (2 marks)
Q10.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 4 and 6?
Your answer: (2 marks)
Section B · LCM and HCF via prime factorisation

Use prime factorisation to find LCM and HCF, including three-number cases.

LEARNING OBJECTIVE 2 · GRADE 4–5

Find LCM and HCF using prime factorisation.

Success criteria — I can:
  • write each number as a product of prime factors
  • for HCF: take the primes that appear in BOTH, each to its SMALLEST power
  • for LCM: take ALL primes that appear, each to its LARGEST power
Q11.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 15 and 18?
Your answer: (3 marks)
Q12.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 14 and 21?
Your answer: (3 marks)
Q13.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 14 and 21?
Your answer: (2 marks)
Q14.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 12 and 16?
Your answer: (3 marks)
Q15.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 24 and 36?
Your answer: (3 marks)
Q16.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 27 and 48?
Your answer: (2 marks)
Q17.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 48 and 60?
Your answer: (3 marks)
Q18.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 72 and 108?
Your answer: (3 marks)
Q19.
What is the LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. of 11, 33, and 44?
Your answer: (3 marks)
Q20.
What is the HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. of 11, 33, and 121?
Your answer: (3 marks)
Section C · Word problems

Apply LCM and HCF to real-life contexts: scheduling, tiling, equal shares.

LEARNING OBJECTIVE 3 · GRADE 4–5

Recognise and solve LCM and HCF word problems.

Success criteria — I can:
  • recognise LCM problems (when do two repeating events next happen together?)
  • recognise HCF problems (biggest equal share / biggest tile / biggest bag)
  • set up and complete the calculation, then check the answer in context
Q21.
Lars goes swimming every 2 days. Rita goes every 3 days. They both went swimming today. After how many days will they next both go swimming on the same day?
Your answer: (2 marks)
Q22.
From Q21: today is Friday. What day of the week will Lars and Rita next swim together?
Your answer:
(2 marks)
Q23.
Rita goes swimming every 3 days. Alan goes every 5 days. They both swam today. After how many days will they next swim together?
Your answer: (2 marks)
Q24.
All three friends from Q21 and Q23 (Lars every 2 days, Rita every 3, Alan every 5) swam together today. After how many days will all three next swim together?
Your answer: (3 marks)
Q25.
From Q24: today is Friday. What day of the week will all three next swim together (30 days from now)?
Your answer:
(3 marks)
Q26.
Two buses leave a station together. Bus A returns every 8 minutes. Bus B returns every 12 minutes. After how many minutes will they next leave together?
Your answer: (3 marks)
Q27.
A floor measures 360 cm by 240 cm. It must be tiled with square tiles of equal size, no gaps, no cutting. What is the largest possible tile size (in cm)?
Your answer: (3 marks)
Q28.
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. = 60. One number is 20. Find the other.
Your answer: (3 marks)
Q29.
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. = 216. Find their LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12..
Your answer: (3 marks)
Q30.
Find the smallest number that is divisible byGoes in exactly, with no remainder. 36 is divisible by 4 because 36 ÷ 4 = 9. 4, 6, and 10.
Your answer: (3 marks)
Q31.
Evan is making sandwiches. Each sandwich needs 1 bread roll, 1 cheese slice, and 1 ham slice. Bread rolls come in packs of 9, cheese in packs of 6, ham in packs of 8. He must finish with no leftovers in any pack. What is the minimum number of sandwiches he could make?
Your answer: (3 marks)
Q32.
Medika has 36 violas, 54 sweet peas, and 72 fuchsias. She wants to plant them in pots so that each pot has only one type of plant AND every pot has the same number of plants. What is the maximum number of plants per pot?
Your answer: (3 marks)
Q33.
From Q32: if Medika puts 18 plants in each pot, how many pots will she need in total?
Your answer: (2 marks)
Section D · Higher reasoning

Use HCF and LCM as tools in cross-topic problems.

LEARNING OBJECTIVE 4 · GRADE 6

Reason with the structure of HCF and LCM at Higher level.

Success criteria — I can:
  • use the identity HCF × LCM = a × b to find unknowns
  • find one number given HCF, LCM and the other number
  • apply HCF and LCM in cross-topic problems (fractions, ratio, geometry)
Q34.
Simplify the fraction 168/252 to its lowest terms. Type the numerator in the first box and the denominator in the second.
Your answer:
,
(3 marks)
Q35.
Calculate 1/12 + 1/18. Give your answer as a single fraction in lowest terms. Type the numerator in the first box and the denominator in the second.
Your answer:
,
(3 marks)
Q36.
A school has 168 boys and 192 girls. Write the ratio boys : girls in its simplest form. Type the boys-part in the first box and the girls-part in the second.
Your answer:
,
(3 marks)
Q37.
A square has a perimeter equal to LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12.(15, 20) cm. What is the area of the square, in cm2?
Your answer: (4 marks)
Q38.
Two whole numbersA number with no fractional part: 0, 1, 2, 3, 4, … a and b have HCFHighest common factor. The biggest number that divides both a and b exactly. HCF(12, 18) = 6. = 12 and LCMLowest common multiple. The smallest number that is a multiple of both a and b. LCM(4, 6) = 12. = 180. Both numbers are greater than 20 and less than 100. Find a + b.
Your answer: (5 marks)
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