Highest common factor

Description

The highest common factor (HCF) of two or more numbers is the largest number that divides exactly into all of them.

Method 1 — list factors: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 HCF = 6

Method 2 — prime factor decomposition: $$12 = 2^2 \times 3 \qquad 18 = 2 \times 3^2$$ Multiply shared prime factors using the lowest power of each: $$\text{HCF}(12, 18) = 2^1 \times 3^1 = 6$$

Using a Venn diagram: place prime factors of each number in a Venn diagram — the HCF is the product of factors in the intersection.

Why it matters: used to simplify fractions (divide numerator and denominator by HCF) and to simplify ratios.

Common error: confusing HCF with LCM — HCF divides into the numbers, LCM is a multiple of the numbers.

Links

Prime factor decomposition Simplifying fractions Lowest Common Multiple