HCF and LCM Relationship

\( ab=\operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b) \)
Number GCSE
Question 11 of 20
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\( \text{Find the LCM of 6 and 10 given HCF=2.} \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( Use LCM=(a×b)/HCF. \)

Explanation

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Statement

The Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two integers are connected by a useful relationship. For any two positive integers \(a\) and \(b\):

\[ a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b) \]

Why it’s true

  • Every integer can be expressed as a product of its prime factors.
  • The HCF takes the lowest powers of the primes shared by both numbers.
  • The LCM takes the highest powers of the primes that appear in either number.
  • Multiplying HCF and LCM therefore combines exactly the same prime factors as multiplying the two numbers directly, so the equality holds.

Recipe (how to use it)

  1. Write the two numbers in prime factor form.
  2. Find the HCF by taking the lowest powers of common factors.
  3. Find the LCM by taking the highest powers of all factors.
  4. Check the relationship: multiply HCF and LCM, and confirm it equals \(a \times b\).

Spotting it

This formula is especially useful when you know one of HCF or LCM and need to calculate the other quickly, without recomputing everything.

Common pairings

  • Fractions: simplifying and finding common denominators.
  • Number theory questions with divisibility.
  • Exam problems involving “lcm × hcf” identity.

Mini examples

  1. Given: \(a=12, b=18\). HCF: 6, LCM: 36. Check: \(12 \times 18 = 216\), and \(6 \times 36 = 216\).
  2. Given: \(a=8, b=20\). HCF: 4, LCM: 40. Check: \(8 \times 20 = 160\), and \(4 \times 40 = 160\).

Pitfalls

  • Mixing HCF and LCM: Remember, HCF is the greatest common factor, not the least.
  • Forgetting prime factors: Always break into prime factorisation when unsure.
  • Negative numbers: For this identity, use positive integers.

Exam strategy

  • Use the formula to quickly check answers: if HCF × LCM ≠ \(a \times b\), something went wrong.
  • When given HCF and product of numbers, you can find the LCM directly by dividing: \(\text{LCM} = \frac{a \times b}{\text{HCF}}\).
  • Be careful with large numbers: prime factorisation is more reliable than guessing factors.

Summary

The relationship \(a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)\) links two central concepts of number theory. It saves time in exams, provides a neat way of checking answers, and ensures accuracy when working with multiples and factors.

Worked examples

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  1. Find HCF and LCM of 12 and 18, and check the relationship.
    1. \( 12 = 2^2 × 3 \)
    2. \( 18 = 2 × 3^2 \)
    3. \( HCF = 2 × 3 = 6 \)
    4. \( LCM = 2^2 × 3^2 = 36 \)
    5. \( Check: 12×18=216, 6×36=216 \)
    Answer: \( HCF=6, LCM=36 \)
  2. Find HCF and LCM of 8 and 20, and check the relationship.
    1. \( 8=2^3 \)
    2. \( 20=2^2 × 5 \)
    3. \( HCF=2^2=4 \)
    4. \( LCM=2^3 × 5=40 \)
    5. \( Check: 8×20=160, 4×40=160 \)
    Answer: \( HCF=4, LCM=40 \)
  3. \( For a=15, b=25, verify HCF×LCM=a×b. \)
    1. \( 15=3×5 \)
    2. \( 25=5^2 \)
    3. \( HCF=5 \)
    4. \( LCM=3×5^2=75 \)
    5. \( Check: 15×25=375, 5×75=375 \)
    Answer: \( HCF=5, LCM=75 \)
  4. \( Find LCM of 21 and 28 given HCF=7. \)
    1. \( a×b=21×28=588 \)
    2. \( LCM=(a×b)/HCF=588/7=84 \)
    Answer: 84
  5. \( Find HCF of 36 and 60 given LCM=180. \)
    1. \( a×b=36×60=2160 \)
    2. \( HCF=(a×b)/LCM=2160/180=12 \)
    Answer: 12
  6. \( Check the relationship for a=14, b=35. \)
    1. \( 14=2×7 \)
    2. \( 35=5×7 \)
    3. \( HCF=7 \)
    4. \( LCM=2×5×7=70 \)
    5. \( Check: 14×35=490, 7×70=490 \)
    Answer: \( HCF=7, LCM=70 \)
  7. \( Given HCF=3, LCM=84, and one number=12, find the other number. \)
    1. \( Use formula: ab=HCF×LCM=3×84=252 \)
    2. \( a=12, so b=252/12=21 \)
    Answer: 21
  8. \( Find LCM of 9 and 10 using HCF=1. \)
    1. \( a×b=9×10=90 \)
    2. \( LCM=90/1=90 \)
    Answer: 90
  9. \( Verify the identity for a=16, b=24. \)
    1. \( 16=2^4 \)
    2. \( 24=2^3×3 \)
    3. \( HCF=2^3=8 \)
    4. \( LCM=2^4×3=48 \)
    5. \( Check: 16×24=384, 8×48=384 \)
    Answer: \( HCF=8, LCM=48 \)
  10. Given numbers 30 and 45, find HCF, LCM and verify.
    1. \( 30=2×3×5 \)
    2. \( 45=3^2×5 \)
    3. \( HCF=3×5=15 \)
    4. \( LCM=2×3^2×5=90 \)
    5. \( Check: 30×45=1350, 15×90=1350 \)
    Answer: \( HCF=15, LCM=90 \)