The HCF in mathematics stands for `highest common factor`, also known as `greatest common divisor (gcd)`. The HCF is the largest number that can evenly divide two or more numbers. It's useful when simplifying fractions or reducing ratios, and is closely related to the [[lcm]] and [[lcd]]. Example: > [!NOTE] > Problem: `hcf(12, 18)` > Factors of 12: 1, 2, 3, 4, 6, 12 > Factors of 18: 1, 2, 3, 6, 9, 18 > Common Factors: 1, 2, 3, 6 > Highest Common Factor: `6` ## Calculating HCF The HCF can be calculated by listing all the factors of each number and picking the largest one they have in common. While this is easy for small numbers, it gets impractical for larger ones. **Calculating via Prime Factorisation:** Break down each number into its [[prime factors]]. Then, identify the prime factors they have **in common** and multiply the **lowest powers** of those primes. This is similar to both [[lcm]] & [[lcd]] as they are all closely linked. **Two Numbers:** > [!TIP] > Find `HCF(18, 24)` > > - $18 = 2 \cdot 3^2$ > > - $24 = 2^3 \cdot 3$ > Common primes: $2$ and $3$ > Take the **lowest powers**: $2^1$, $3^1$ > **HCF = $2 \cdot 3 = 6$** > **Three Numbers:** > [!TIP] > Find `HCF(36, 60, 72)` > > - $36 = 2^2 \cdot 3^2$ > > - $60 = 2^2 \cdot 3 \cdot 5$ > > - $72 = 2^3 \cdot 3^2$ > Common primes: $2$ and $3$ > Take the **lowest powers**: $2^2$, $3^1$ > **HCF = $4 \cdot 3 = 12$** > > [!NOTE] > The [[hcf]] and [[lcm]] are closely linked: > $a \cdot b = \text{hcf}(a, b) \cdot \text{lcm}(a, b)$ > see also: [[hcf-lcm product identity]] ## Practice Tests - [[mcq-on-lcm-and-hcf-5eea6a1039140f30f369e847-9aefe4b0.pdf]]