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Hcf Lcm

Hcf Lcm

HCF and LCM problems

Advanced Hcf Lcm Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. HCF or LCM?
  4. Working backwards from the HCF and LCM
  5. HCF and LCM of algebraic terms
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.1: HCF and LCM problems
  • 1.1Decide whether a word problem needs the HCF or the LCM, and solve it.
  • 1.1Work backwards from a given HCF and LCM.
  • 1.1Find the HCF and LCM of algebraic terms.

Before you start: prime factors, HCF and LCM.

Formula bank

Use the LCMLearn ithappen together again; same number of each; smallest number of packsthe answer is BIGGER than the numbers.Use it when cycles, packs.
Use the HCFLearn itequal pieces as long as possible; largest tiles; most identical groupsthe answer is SMALLER than the numbers.Use it when cutting, sharing, tiling.
Numbers from HCF and LCMLearn itnumbers \(ha\), \(hb\) with \(ab=\dfrac{\text{LCM}}{\text{HCF}}\), \(a,b\) sharing no factorUse it when "find the two numbers".
Algebraic termsLearn itletters behave like primes: HCF lower powers, LCM higher powersHCF of \(12a^{3}b\) and \(20a^{2}b^{2}\) is \(4a^{2}b\).Use it when terms with letters.

HCF or LCM?

General methodHow to answer any HCF / LCM word problem

  1. Ask: will the answer be BIGGER than the numbers (a common multiple: LCM) or SMALLER (something that fits into each: HCF)?
  2. Find it (prime factors or listing).
  3. Use it to answer the actual question (a time, a number of packs, a number of pieces).

Example A green light flashes every 12 minutes, a red one every 45 minutes. They flash together at 9 am. When next together?

A time both cycles reach: the LCM.

  1. \(12=2^{2}\times3\), \(45=3^{2}\times5\)
  2. LCM = 180 min = 3 h

Answer 12 noon

Example Ribbons of 84 cm and 126 cm are cut into equal pieces, as long as possible, none left over. How many pieces?

A length that fits into both: the HCF.

  1. HCF(84, 126) = 42 cm
  2. 84 ÷ 42 + 126 ÷ 42 = 2 + 3

Answer 5 pieces

Try it yourself

1
Two bells ring every 4 and 6 minutes. Rung together now, next together in how many minutes?
Show answer

Answer 12

2
A floor 360 cm by 225 cm is covered with the largest identical square tiles, none cut. How many tiles?
Hint
HCF = 45 cm.
Show answer

Answer 40

3
Packs of 6, 5 and 12; the same number of each fruit. How many packs of the 6-pack?
Hint
LCM = 60.
Show answer

Answer 10

Working backwards from the HCF and LCM

General methodHow to find two numbers from their HCF and LCM

  1. Write them as \(ha\) and \(hb\) (h = HCF).
  2. \(ab=\text{LCM}\div\text{HCF}\); list pairs \(a,b\) that share no factor.
  3. Use any extra condition (e.g. both > 20) to choose.

Example Two numbers > 20 have HCF 12 and LCM 180. Find them.

\(ab=180\div12=15\): pairs (1, 15) and (3, 5).

  1. (1, 15) → 12, 180 (12 is not > 20)
  2. (3, 5) → 36, 60

Answer 36 and 60

Try it yourself

1
Two numbers have HCF 6 and LCM 36. One is 12. The other?
Show answer

Answer 18

HCF and LCM of algebraic terms

General methodHow to find the HCF and LCM of algebraic terms

  1. Numbers: HCF / LCM as usual.
  2. Each letter: HCF uses the LOWER power of letters in both; LCM uses the HIGHER power of every letter.

Example HCF and LCM of \(12a^{3}b\) and \(20a^{2}b^{2}\).

Numbers, then each letter.

  1. HCF: \(4a^{2}b\)
  2. LCM: \(60a^{3}b^{2}\)

Answer \(4a^{2}b\) and \(60a^{3}b^{2}\)

Try it yourself

1
HCF of \(15x^{2}y^{3}\) and \(25x^{4}y\)?
Show answer

Answer \(5x^{2}y\)

2
LCM of \(15x^{2}y^{3}\) and \(25x^{4}y\)?
Show answer

Answer \(75x^{4}y^{3}\)

Ways they test it

Type 1 Together again (LCM)

"Buses leave every 15 and 25 minutes, both at 7.40 am. When next together?"

Spot it repeating cycles.

  1. LCM of the cycles.
  2. Add to the start time.

For example 15 and 25 min from 7.40

  1. LCM = 75 min

Answer 8.55 am

Type 2 Same number of each (LCM)

"Rulers in 6s, pens in 8s, erasers in 15s. Smallest number of packs of each?" (3 marks)

Spot it packs of different sizes.

  1. LCM of the pack sizes.
  2. ÷ each pack size.

For example 6, 8, 15

  1. LCM = 120

Answer 20, 15 and 8 packs

Type 3 Equal pieces / tiles (HCF)

"A 480 cm by 270 cm floor with the largest square tiles. How many tiles?"

Spot it "as large as possible", "no waste".

  1. HCF.
  2. Count along each side, multiply.

For example 480 by 270

  1. HCF = 30
  2. 16 × 9

Answer 144 tiles

Type 4 Two numbers from HCF and LCM

"HCF 18, LCM 216, both > 20. Find them."

Spot it HCF and LCM given.

  1. \(ab=\text{LCM}\div\text{HCF}\).
  2. Coprime pairs, then the condition.

Type 5 Algebra

"\(P=x^{2}y^{n}\), \(Q=x^{n-1}y^{4}\), \(n>4\). HCF?"

Spot it letters or powers in n.

  1. Compare powers of each letter; take the lower.

Mistakes to avoid

  • ✗ Using the HCF for 'together again'✓ LCMThe answer must be a multiple of both.
  • ✗ Stopping at the HCF length✓ answer the question (number of pieces)Read the last line.
  • ✗ LCM = the numbers multiplied✓ usually smallerOnly when the HCF is 1.
  • ✗ HCF of letters with the higher power✓ the lower powerIt must divide both terms.

Summary

  • Together again / same number of each → LCM (answer bigger).
  • Equal pieces / largest tiles / groups → HCF (answer smaller).
  • From HCF h and LCM L: \(ha\), \(hb\) with \(ab=L\div h\).
  • Letters behave like primes.

If you see …

  • "next time together"LCM.
  • "as long / large as possible"HCF.

Practice

Practice: easy, medium, hard

Easy

1
Cut 90 cm and 60 cm into equal pieces, as long as possible. Piece length (cm)?
Show working
  1. HCF(90, 60)

Answer 30

2
Pens in packs of 8, 10 and 12, the same number of each. Smallest number of pens of each?
Show working
  1. LCM(8, 10, 12)

Answer 120

3
LCM of 20 and 30?
Show working
  1. 20, 40, 60

Answer 60

Medium

1
Ribbons of 72 cm and 96 cm cut into equal pieces, as long as possible. How many pieces?
Guide me

What is it asking? number of pieces

What do we know? pieces fit both

Plan HCF, then divide

Show working
  1. HCF = 24
  2. 3 + 4 = 7

Answer 7

2
Buses leave every 15 and 25 minutes. Both leave at 7.40 am. Next time together?
Guide me

What is it asking? a time

What do we know? two cycles

Plan LCM, add to 7.40

Show working
  1. LCM = 75 min
  2. 7.40 + 1 h 15 min

Answer 8.55 am

3
A floor 480 cm by 270 cm is covered with the largest square tiles possible. How many tiles?
Guide me

What is it asking? number of tiles

What do we know? tile side divides both

Plan HCF, then count

Show working
  1. HCF = 30
  2. \(16\times9\)

Answer 144

Hard

1
Rulers in packs of 6, pens in 8s, erasers in 15s; the same number of each. Smallest number of packs of pens?
Guide me

What is it asking? packs of pens

What do we know? LCM of 6, 8, 15

Plan LCM ÷ 8

Show working
  1. LCM = 120
  2. \(120\div8\)

Answer 15

PartAnswer / workingMarksGuidance
120M1
15A1
Total: 2 marks
2
Two numbers, both > 20, have HCF 18 and LCM 216. Find them.
Guide me

What is it asking? two numbers

What do we know? \(ab=216\div18=12\)

Plan coprime pairs: (1, 12), (3, 4)

Show working
  1. (1, 12) → 18, 216
  2. (3, 4) → 54, 72

Answer 54 and 72

PartAnswer / workingMarksGuidance
ab = 12M1
54 and 72A1
Total: 2 marks
3
\(P=x^{2}y^{n}\), \(Q=x^{n-1}y^{4}\), x and y prime, \(n>4\). HCF of P and Q?
Guide me

What is it asking? the HCF

What do we know? compare powers

Plan x: 2 vs n − 1; y: n vs 4

Show working
  1. n − 1 > 3, so lower x power is 2
  2. lower y power is 4

Answer \(x^{2}y^{4}\)

PartAnswer / workingMarksGuidance
\(x^{2}y^{4}\)B2
Total: 2 marks

Can I do this?

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

Quiz

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