Elite Learners

Home›11+›Maths›Number and place value›HCF and LCM mastery

HCF and LCM mastery

Highest common factors and lowest common multiples in multi-step puzzles.

Video coming soon The written summary and downloads below are complete and ready to use.

In Lesson 1.5 you found the factors and multiples of a number. Now you will compare two numbers and find what they have in common.

In this lesson you will learn:

  • how to find the highest common factor (HCF) of two numbers
  • how to find the lowest common multiple (LCM) of two numbers
  • how to tell which one a question needs
  • how to use them to solve puzzles in more than one step

1. Common factors and the HCF

Key idea A common factor is a factor of both numbers. The highest common factor (HCF) is the biggest one.

How to find the HCF

  1. List the factors of each number
    Use factor pairs, as in Lesson 1.5, so you don’t miss any.
  2. Circle the numbers that are in both lists
    These are the common factors.
  3. The biggest one is the HCF

Example: the HCF of 12 and 18

Factors of 121234612

Factors of 181236918

  • The common factors are 1, 2, 3 and 6.
  • The biggest is 6.
AnswerHCF of 12 and 18 = 6

2. Common multiples and the LCM

Key idea A common multiple is in both times tables. The lowest common multiple (LCM) is the smallest one.

How to find the LCM

  1. Count up in the times table of the bigger number
  2. Check each one: is it in the other times table too?
  3. The first one that is in both is the LCM

Example: the LCM of 4 and 6

Multiples of 66121824

  • 6 is not in the 4 times table. 12 is: 4 × 3 = 12.
AnswerLCM of 4 and 6 = 12
Watch out Multiplying the two numbers always gives a common multiple, but not always the lowest. 4 × 6 = 24, but the LCM of 4 and 6 is 12.

3. HCF or LCM? Spot the clues

Key idea HCF: splitting things into the biggest equal groups or pieces. LCM: things that repeat and happen together again.
Clue in the questionYou needBecause
“largest”, “greatest”, “cut into equal pieces”, “share into identical groups”HCFThe answer is smaller than (or equal to) the numbers
“at the same time”, “together again”, “smallest number of each”, “next time both”LCMThe answer is bigger than (or equal to) the numbers
Two handy shortcuts
  • If one number is a factor of the other, like 4 and 12: the HCF is the smaller one (4) and the LCM is the bigger one (12).
  • If the only common factor is 1, like 4 and 9: the HCF is 1 and the LCM is the two numbers multiplied: 4 × 9 = 36.

4. Puzzles in more than one step

Key idea Find the HCF or LCM first. Then read the question again: it often asks one more thing.

Example 1: tiling a floor

A floor is 12 m long and 18 m wide. It is covered with square tiles, all the same size, with no gaps or cutting. What is the biggest tile that can be used, and how many are needed?

  1. The tile must fit into 12 and into 18 exactly: that is a common factor
  2. The biggest tile is the HCF of 12 and 18: 6 m
  3. 12 ÷ 6 = 2 tiles along, 18 ÷ 6 = 3 tiles across. 2 × 3 = 6 tiles
Answer6 m tiles, and 6 of them

Example 2: buses

The red bus leaves every 12 minutes and the blue bus every 15 minutes. They both leave at 9:00. When do they next leave together?

  1. “Together again” means the LCM of 12 and 15
    Count up in 15s: 15, 30, 45, 60. 60 is in the 12 times table.
  2. 60 minutes after 9:00 is 10:00
Answer10:00

Example 3: sausages and buns

Sausages come in packs of 8. Buns come in packs of 6. Tom wants the same number of each. What is the smallest number of packs of each he can buy?

  1. The same number of each: the LCM of 8 and 6 is 24
  2. 24 ÷ 8 = 3 packs of sausages, 24 ÷ 6 = 4 packs of buns
Answer3 packs of sausages and 4 packs of buns

The mistakes that cost marks

  1. Mixing up HCF and LCM. remember the HCF is never bigger than the numbers, and the LCM is never smaller.
  2. Missing a factor in the lists. use factor pairs so the lists are complete before you compare them.
  3. Just multiplying the two numbers for the LCM. 4 × 6 = 24, but the LCM is 12. count up the bigger number’s times table and stop at the first one in both.
  4. Forgetting that the HCF can be one of the numbers. The HCF of 4 and 12 is 4. check whether the smaller number goes into the bigger one.
  5. Stopping too soon in a puzzle. The HCF of 12 and 18 is 6, but the question asked how many tiles. read the question again and do the last step.

Quick checklist

Tick each one when you can do it without help.

Download

Print these and work through them away from the screen.

Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

All of this is free

Every 11+ maths resource on this site is free, with no account needed. Taught sessions are currently GCSE maths only — if you would like a group opened for this subject, join the waiting list and it helps me decide what to run next.

Join the waiting list →