Number operations and integers Study Guide

The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are fundamental concepts in number theory. The HCF of two or more numbers is the largest positive integer that divides each of the integers without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6. The largest of these is 6, so the HCF of 12 and 18 is 6.

To find the HCF of two numbers, one method is to list all factors of each number and identify the largest factor they share. For larger numbers, prime factorisation is more efficient. Write each number as a product of its prime factors. The HCF is found by multiplying all the common prime factors (raised to the lowest power they appear in either factorisation).

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the integers. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, ... The multiples of 5 are 5, 10, 15, 20, ... The smallest number that appears in both lists is 15, so the LCM of 3 and 5 is 15.

Similar to HCF, for larger numbers, prime factorisation is very useful for finding the LCM. Write each number as a product of its prime factors. The LCM is found by multiplying all prime factors that appear in any of the factorisations (raised to the highest power they appear in either factorisation).

A useful relationship exists for two numbers, 'a' and 'b': HCF(a, b) × LCM(a, b) = a × b. This formula can help check your answers or find one value if the others are known. Both HCF and LCM have practical applications, such as in simplifying fractions (using HCF) or solving problems involving cycles or periodic events (using LCM).

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides into all of them without leaving a remainder. For example, to find the HCF of 12 and 18, we list their factors: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 The common factors are 1, 2, 3, 6. The highest among these is 6, so HCF(12, 18) = 6.

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all of them. For example, to find the LCM of 4 and 6, we list their multiples: Multiples of 4: 4, 8, 12, 16, 20, 24, ... Multiples of 6: 6, 12, 18, 24, 30, ... The common multiples are 12, 24, ... The smallest among these is 12, so LCM(4, 6) = 12.

Another way to find HCF and LCM is by using prime factorization. First, find the prime factorization of each number. For HCF, multiply all the common prime factors, raised to the lowest power they appear in any of the factorizations. For LCM, multiply all prime factors (common and uncommon), raised to the highest power they appear in any of the factorizations. For example: For 12 and 18: 12 = 2² × 3 18 = 2 × 3² HCF(12, 18) = 2¹ × 3¹ = 6 (lowest power of common primes) LCM(12, 18) = 2² × 3² = 4 × 9 = 36 (highest power of all primes)