Highest common factor (HCF)

Understanding the Highest Common Factor (HCF)

  • The Highest Common Factor (HCF), also known as the greatest common divisor (GCD), is the largest number that divides exactly into two or more numbers.
  • It is one of the fundamental concepts in algebra that is important for solving algebraic fractions, simplifying expressions and solving equations.
  • The HCF indicates what to ‘take out’ as a common factor when simplifying an expression and is key when fraction simplifying.

Finding the HCF of Numbers

  • The HCF of two or more numbers can be found by listing all the factors of each number and choosing the largest one that’s common to all.
  • Alternatively, the HCF can be found by repeatedly dividing the lot by a prime number, known as prime factorization, until no further division is possible. The HCF is the product of the smallest power of the common prime factors.
  • Using a VDU (Venn Diagram Understanding) method also helps to visualise the HCF.

HCF and Algebra

  • The HCF concept applies to algebraic terms as well as numeric. For example, the HCF of 18x^3 and 12x^2 is 6x^2.
  • The HCF of algebraic expressions can also be found by prime factorization, just like numbers.
  • For algebraic expressions, the HCF is determined by identifying the smallest power of each common variable, as well as any numeric factors.

HCF in Solving Equations

  • Knowing the HCF of algebraic terms within an equation can greatly reduce its complexity and makes it easier to solve.
  • An equation has to be ‘factored out’ using HCF before it can be solved.
  • For example, consider the equation 2x^2 = 2x. We take out a factor of 2x from both sides to get 2x(x - 1) = 0. From there, it’s easy to see that the solutions are x = 0 and x = 1.

Mastering HCF

  • It’s crucial to be proficient with finding the HCF to facilitate algebraic manipulations.
  • Regular practice with different types of algebraic expressions and equations will enhance understanding and speed of applying the concept of HCF.
  • Mastery of HCF is essential for success in algebra, making it an important revision focus.