Percentages & Fractions

How to Find the LCM and GCF (Three Methods)

By Math Solving Space · · 2 min read

On this page
  1. Method 1: Listing
  2. Method 2: Prime factorisation
  3. Method 3: Euclid's algorithm (for the GCF)
  4. The shortcut linking them
  5. Where you use them
  6. Common mistakes
  7. Practice questions
  8. Frequently asked questions

The greatest common factor (GCF) is the largest number that divides all your numbers. The least common multiple (LCM) is the smallest number they all divide into. You can find both by listing, by prime factorisation, or (for the GCF) with Euclid's algorithm. For two numbers, GCF×LCM=a×b\text{GCF} \times \text{LCM} = a \times b.

Method 1: Listing

GCF of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The largest shared factor is 6.

LCM of 12 and 18. Multiples of 12: 12, 24, 36, 48… Multiples of 18: 18, 36, 54… The first shared multiple is 36.

Listing is fine for small numbers but slow for big ones.

Method 2: Prime factorisation

Write each number as a product of primes:

12=22×318=2×3212 = 2^2 \times 3 \qquad 18 = 2 \times 3^2
  • GCF: take each shared prime to its lowest power: 21×31=62^1 \times 3^1 = 6.
  • LCM: take every prime to its highest power: 22×32=362^2 \times 3^2 = 36.

Three numbers: 4=224 = 2^2, 6=2×36 = 2 \times 3, 10=2×510 = 2 \times 5. The LCM is 22×3×5=602^2 \times 3 \times 5 = 60.

Method 3: Euclid's algorithm (for the GCF)

Divide the larger number by the smaller and keep the remainder. Repeat with the smaller number and the remainder until the remainder is 0.

GCF of 48 and 180:

180=3×48+3648=1×36+1236=3×12+0\begin{aligned} 180 &= 3 \times 48 + 36 \\ 48 &= 1 \times 36 + 12 \\ 36 &= 3 \times 12 + 0 \end{aligned}

The last non-zero remainder, 12, is the GCF. This works quickly even for very large numbers.

The shortcut linking them

For two numbers:

LCM(a,b)=a×bGCF(a,b)\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}

So LCM(48,180)=48×18012=720\text{LCM}(48, 180) = \frac{48 \times 180}{12} = 720.

Where you use them

  • GCF: simplifying fractions (48180=415\frac{48}{180} = \frac{4}{15}), splitting things into equal groups.
  • LCM: common denominators for adding fractions, timing repeating events (two buses every 12 and 18 minutes meet every 36 minutes).

Common mistakes

  • Mixing up the two: the GCF is never bigger than the smallest number; the LCM is never smaller than the largest.
  • Using highest powers for the GCF.
  • Stopping Euclid's algorithm one step early.

Practice questions

  1. GCF of 36 and 60.
  2. LCM of 8 and 14.
  3. GCF and LCM of 15, 25 and 40.

Answers: 1) 12 2) 56 3) GCF 5, LCM 600

Check your answers with the LCM calculator and the GCF calculator, which shows Euclid's algorithm step by step. Prime factors come from the prime number calculator, and you will use the LCM when adding fractions.

Frequently asked questions

Are GCF, HCF and GCD the same thing?

Yes. Greatest common factor, highest common factor and greatest common divisor all mean the same.

What if the GCF is 1?

The numbers are coprime. Their LCM is then just their product.

Can I find the LCM of more than two numbers?

Yes. Use prime factorisation with the highest power of each prime, or find the LCM of two numbers, then of that result with the next.

Open the calculator →

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