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Prime Factors, HCF and LCM

Number

What each one means

  • Where a single value divides two numbers exactly, it is a common factor of the pair, and 1 always qualifies
  • The highest common factor, or HCF, is the largest such value
  • A value sitting in both numbers' times tables is a common multiple
  • Of the infinitely many such values, the smallest is the lowest common multiple, or LCM
  • The HCF is what cancels a fraction to its simplest form; the LCM is what supplies a common denominator

Finding them by listing

  • List every factor of both numbers, then take the largest value common to the two lists
  • List the multiples of both, then take the smallest value common to the two lists
  • Listing is quick for small numbers and unreliable for large ones

Finding them from prime factors

  • Decompose both numbers into primes first
  • Place the primes the two numbers share in the overlap of a Venn diagram, taking each shared prime only as many times as it appears in both
  • Place the remaining primes in the outer region belonging to their own number
  • Multiplying the overlap gives the HCF; multiplying everything in the whole diagram gives the LCM
Venn diagram for the prime factors of 42 and 90: 7 sits in the region for 42 alone, 3 and 5 sit in the region for 90 alone, and 2 and 3 sit in the overlap shared by both
Source: Highest common factor (HCF) by Save My Exams
  • Where no primes are shared, the overlap holds a 1 and the HCF is 1
Worked example

Finding the HCF and LCM from prime factors

Find the highest common factor and the lowest common multiple of 60 and 144.

Solution:

  • Decompose each into primes
  • 60 = 2² × 3 × 5
  • 144 = 2⁴ × 3²
  • Both contain 2 and 3. Take the lower index of each for the shared part: 2² and 3
  • HCF = 2² × 3 = 12
  • For the LCM collect every prime occurring anywhere, each at its highest index: 2⁴, 3² and 5
  • LCM = 16 × 9 × 5 = 720