HCF and LCM

HCF and LCM

Highest Common Factor (HCF)

  • The Highest Common Factor (HCF) also known as the greatest common divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

  • To find the HCF of two numbers, write out all the factors of the two numbers and identify the largest common factor between them. For example, the HCF of 12 and 18 is 6.

  • There’s also a method of finding the HCF using prime factorisation. Break down each of the numbers into their prime factors and multiply the lowest powers of common factors. For instance, the HCF of 48 (2^4 × 3) and 60 (2^2 × 3 × 5) is 2^2 × 3 = 12.

  • An efficient method to find the HCF is by using the Euclidean algorithm. It is a method of repeatedly subtracting the smaller number from the larger until you reach a remainder of 0. The last non-zero remainder is the HCF of the two numbers.

Least Common Multiple (LCM)

  • The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is perfectly divisible by each of the numbers.

  • One way to find the LCM is to list the multiples of the given numbers and to identify the smallest common multiple. For example, the LCM of 4 and 6 is 12.

  • The LCM can also be found by using prime factorisation. Write down the numbers as a product of their prime factors, then multiply the highest power of all factors. For example, the LCM of 12 (2^2 × 3) and 18 (2 × 3^2) is 2^2 × 3^2 = 36.

  • The LCM of two numbers can also be found using the formula: LCM(a, b) = | a × b | / gcd(a, b), where gcd(a, b) is the greatest common divisor of a and b, or their HCF.

Real-Life Applications

  • The concepts of HCF and LCM are used in numerous real-life situations, such as working out the least amount of time until an event recurs, or when reducing fractions to their simplest form.

  • HCF is used when dealing with shared resources, for example, the greatest number of identical cupcakes that can be made from the flour and sugar available.

  • LCM is utilised when you need to find the smallest quantity that can be divided evenly into, for instance, to find the smallest number of identical boxes that can contain a fixed amount of items without any space being wasted.