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Hcf Lcm

Hcf Lcm

Prime factors, HCF and LCM

Simple Hcf Lcm Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Prime factorisation
  4. HCF and LCM
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 1.1: prime factors, HCF and LCM
  • 1.1Write a number as a product of prime factors in index form.
  • 1.1Find the HCF and LCM of two or three numbers.
  • 1.1Use the prime factors to find the largest odd common factor and to multiply factorisations.

Before you start: prime numbers, factors and multiples.

Formula bank

Prime factorisationLearn ite.g. \(360=2^{3}\times3^{2}\times5\)every whole number above 1 has exactly one.Use it when "product of prime factors".
HCFLearn itprimes in BOTH numbers, each to the LOWER powerthe biggest number dividing both.Use it when "highest common factor", sharing into equal groups.
LCMLearn itevery prime in EITHER number, each to the HIGHER powerthe smallest number both divide into.Use it when "lowest common multiple", things happening together.
CheckLearn it\(\text{HCF}\times\text{LCM}=a\times b\)for two numbers.Use it when checking your answers.

Prime factorisation

General methodHow to write any number as a product of prime factors

  1. Divide by the smallest prime that goes exactly (2, then 3, 5, 7, …).
  2. Keep dividing the result until you reach 1 (a 'ladder'), or split into factor pairs until every branch is prime (a 'tree').
  3. Write the primes multiplied, in index form.

Example Write 360 as a product of prime factors.

Divide by the smallest prime each time (a division ladder).

  1. 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45
  2. 45 ÷ 3 = 15, ÷ 3 = 5, ÷ 5 = 1
  3. primes used: 2, 2, 2, 3, 3, 5

Answer \(2^{3}\times3^{2}\times5\)

Try it yourself

1
Write 252 as a product of prime factors.
Show answer

Answer \(2^{2}\times3^{2}\times7\)

2
Write 588 as a product of prime factors.
Show answer

Answer \(2^{2}\times3\times7^{2}\)

3
Write 72 in index form.
Show answer

Answer \(2^{3}\times3^{2}\)

HCF and LCM

The HCF divides both numbers, so it can only use primes they SHARE, each to the lower power. The LCM must contain both numbers, so it uses every prime, each to the higher power.

Prime factors of 252 and 58825258832 23 77
Shared primes go in the overlap.

HCF = the overlap: \(2\times2\times3\times7=84\).

LCM = everything: \(3\times2\times2\times3\times7\times7=1764\).

General methodHow to find the HCF and LCM of any numbers

  1. Write each number as a product of prime factors.
  2. HCF: take each prime that appears in ALL the numbers, with its LOWEST power; multiply.
  3. LCM: take every prime that appears in ANY number, with its HIGHEST power; multiply.
  4. Small numbers? Listing factors and multiples works too.

Example Find the HCF and LCM of 84 and 120.

Prime factors, then lower / higher powers.

  1. \(84=2^{2}\times3\times7\), \(120=2^{3}\times3\times5\)
  2. HCF \(=2^{2}\times3=12\)
  3. LCM \(=2^{3}\times3\times5\times7=840\)

Answer 12 and 840

Try it yourself

1
HCF of 36 and 54?
Show answer

Answer 18

2
LCM of 8 and 12?
Show answer

Answer 24

3
HCF of 48 and 80?
Show answer

Answer 16

General methodHow to find the largest odd common factor

  1. Find the HCF.
  2. Remove every factor 2 from it (divide by 2 until it is odd).

Example Largest odd number that is a factor of 84 and of 126?

HCF, then remove the 2s.

  1. HCF \(=2\times3\times7=42\)
  2. remove the 2: \(3\times7\)

Answer 21

Ways they test it

Type 1 Product of prime factors

"Write 390 as a product of its prime factors." (2 marks)

Spot it "product of prime factors".

  1. Ladder or tree (M1).
  2. Primes multiplied, index form (A1).

For example 390

  1. ÷ 2 = 195, ÷ 3 = 65, ÷ 5 = 13

Answer \(2\times3\times5\times13\)

Type 2 HCF / LCM from given factorisations

"\(300=2^{2}\times3\times5^{2}\). Find the HCF and LCM of 390 and 300."

Spot it factorisations given.

  1. Line up the primes.
  2. Lower powers for HCF, higher for LCM.

For example \(2\times3\times5\times13\) and \(2^{2}\times3\times5^{2}\)

  1. HCF \(2\times3\times5\)
  2. LCM \(2^{2}\times3\times5^{2}\times13\)

Answer 30 and 3900

Type 3 Largest odd common factor

"Find the largest odd number that is a factor of 96 and 144."

Spot it "odd" with common factor.

  1. HCF.
  2. Remove all 2s.

For example \(96=2^{5}\times3\), \(144=2^{4}\times3^{2}\)

  1. HCF \(=2^{4}\times3=48\)

Answer 3

Type 4 Multiply factorisations

"Write \(360\times588\) as a product of prime factors."

Spot it a product of two factorised numbers.

  1. Add the powers of each prime.

Mistakes to avoid

  • ✗ \(6\times5\times13\)✓ \(2\times3\times5\times13\)6 isn't prime.
  • ✗ HCF with the higher powers✓ lower powersIt must divide both.
  • ✗ LCM = the two numbers multiplied✓ usually smallerOnly when the HCF is 1.
  • ✗ Listing the primes 2, 3, 5, 13✓ multiply them"Product" means ×.

Summary

  • Prime factorisation: divide by the smallest prime until you reach 1; write in index form.
  • HCF: shared primes, lower powers. LCM: all primes, higher powers.
  • HCF × LCM = the two numbers multiplied.
  • Largest odd common factor: HCF without its 2s.

If you see …

  • "product of prime factors"ladder, then index form.
  • "highest common factor"the overlap.
  • "lowest common multiple"everything.

Practice

Practice: easy, medium, hard

Easy

1
Write 72 as a product of prime factors.
Show working
  1. 72 ÷ 2 ÷ 2 ÷ 2 = 9 = 3 × 3

Answer \(2^{3}\times3^{2}\)

2
LCM of 6 and 15?
Show working
  1. 6, 12, 18, 24, 30

Answer 30

3
Largest odd common factor of 60 and 90?
Show working
  1. HCF 30
  2. without the 2: 15

Answer 15

Medium

1
Write 390 as a product of prime factors.
Guide me

What is it asking? the factorisation

What do we know? 390

Plan ladder

Show working
  1. ÷ 2 = 195, ÷ 3 = 65, ÷ 5 = 13

Answer \(2\times3\times5\times13\)

2
\(300=2^{2}\times3\times5^{2}\) and \(390=2\times3\times5\times13\). HCF?
Guide me

What is it asking? the HCF

What do we know? shared 2, 3, 5

Plan lower powers

Show working
  1. \(2\times3\times5\)

Answer 30

3
Same numbers: LCM?
Guide me

What is it asking? the LCM

What do we know? primes 2, 3, 5, 13

Plan higher powers

Show working
  1. \(2^{2}\times3\times5^{2}\times13\)

Answer 3900

Hard

1
Write 420 as a product of prime factors.
Guide me

What is it asking? the factorisation

What do we know? 420

Plan ladder

Show working
  1. ÷ 2 = 210, ÷ 2 = 105, ÷ 3 = 35, ÷ 5 = 7

Answer \(2^{2}\times3\times5\times7\)

PartAnswer / workingMarksGuidance
a correct ladder or treeM1
\(2^{2}\times3\times5\times7\)A1
Total: 2 marks
2
\(420=2^{2}\times3\times5\times7\) and \(1512=2^{3}\times3^{3}\times7\). LCM?
Guide me

What is it asking? the LCM

What do we know? primes 2, 3, 5, 7

Plan higher powers

Show working
  1. \(2^{3}\times3^{3}\times5\times7\)

Answer 7560

PartAnswer / workingMarksGuidance
\(2^{3}\times3^{3}\times5\times7\)M1
7560A1
Total: 2 marks
3
Largest odd number that is a factor of both 96 and 144?
Guide me

What is it asking? an odd common factor

What do we know? HCF 48

Plan remove the 2s

Show working
  1. \(48=2^{4}\times3\)

Answer 3

PartAnswer / workingMarksGuidance
48M1
3A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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