HCF and GCF calculator with steps (highest common factor)

1. Write each number as a product of prime factors: $48 = 2^4 \times 3$ and $180 = 2^2 \times 3^2 \times 5$.
2. Keep only the primes that appear in every number: 2 and 3.
3. Take each with its lowest power: $2^2$ and $3$.
4. Multiply: $\text{HCF} = 2^2 \times 3 = 12$.

If no prime appears in every number, the HCF is 1 and the numbers are called co-prime.

For 48 and 180, put the shared prime factors in the overlap of two circles: $2, 2, 3$. The factors only in 48 ($2, 2$) go on the left; the factors only in 180 ($3, 5$) go on the right.

• HCF = product of the overlap = $2 \times 2 \times 3 = 12$.
• LCM = product of everything in the diagram = $2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 720$.

The calculator draws this diagram whenever you enter two numbers.

• Simplifying fractions: divide the top and bottom by their HCF in one step: $\tfrac{48}{180} = \tfrac{4}{15}$. See how to simplify fractions.
• Factorising: $12x + 18 = 6(2x + 3)$ uses the HCF of 12 and 18.
• Sharing problems: the largest equal groups you can make from 48 red and 180 blue counters, with none left over, is 12 groups.

• Using the highest powers. That gives the LCM.
• Including a prime that isn't in every number. The 5 in 180 isn't a factor of 48, so it's left out.
• Stopping at a common factor that isn't the highest, like 6 for 48 and 180. Prime factors make sure you find the largest.

For two numbers you can use Euclid's algorithm: divide the larger by the smaller and keep the remainder. $180 = 3 \times 48 + 36$, then $48 = 1 \times 36 + 12$, then $36 = 3 \times 12 + 0$. The last non-zero remainder, 12, is the HCF. More in our guide on how to find the HCF.

For three or more numbers, the method is the same: a prime goes into the HCF only if it's in every number, at the lowest power that appears. In the last row, 7 is only in 105, so it's left out, and 5 appears once in 105, so the HCF takes $5^1$. Enter all three numbers in the calculator above to see the factor trees.