Factors, Multiples, LCM and HCF

Introduction to Factors, Multiples, LCM and HCF

Factors and multiples are two fundamental concepts in number theory that describe relationships between whole numbers. Understanding factors and multiples leads naturally to two very useful concepts: the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM), both of which are widely used in simplifying fractions, solving problems involving sharing and grouping, and working with ratios and proportions.

Meaning of a Factor

A factor of a number is any whole number that divides into it exactly, without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these numbers divides exactly into 12. Every whole number greater than 1 has at least two factors: 1 and itself. Numbers with exactly two factors (1 and itself) are called prime numbers, such as 2, 3, 5, 7, 11, and 13, while numbers with more than two factors are called composite numbers, such as 4, 6, 8, 9, and 12.

Finding All Factors of a Number

To find all the factors of a number, systematically test each whole number starting from 1 to see if it divides evenly into the given number, listing pairs of factors as they are found. For example, to find the factors of 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6, and since 5 does not divide evenly, we stop once the factor pairs begin to repeat. This gives the complete list of factors: 1, 2, 3, 4, 6, 8, 12, and 24.

Prime Factorisation

Prime factorisation is the process of expressing a composite number as a product of its prime factors. This is often done using a factor tree, repeatedly breaking a number down into smaller factors until only prime numbers remain. For example, the prime factorisation of 60 is 2 × 2 × 3 × 5, or written using indices, 2² × 3 × 5. Prime factorisation is an essential tool for finding both the HCF and LCM of numbers.

Meaning of a Multiple

A multiple of a number is the result of multiplying that number by any whole number (1, 2, 3, and so on). For example, the multiples of 6 are 6, 12, 18, 24, 30, and so on, continuing indefinitely. Every whole number has an infinite number of multiples, since it can always be multiplied by a larger whole number to produce a new, larger multiple.

Meaning of the Highest Common Factor (HCF)

The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), of two or more numbers is the largest number that divides exactly into all of them. For example, to find the HCF of 12 and 18: the factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 18 are 1, 2, 3, 6, 9, 18; the common factors are 1, 2, 3, and 6, so the HCF is 6, the largest of these common factors.

Finding HCF Using Prime Factorisation

A more efficient method for finding the HCF of larger numbers is to use prime factorisation: express each number as a product of its prime factors, then multiply together the common prime factors, using the lowest power of each shared prime. For example, to find the HCF of 36 (2² × 3²) and 60 (2² × 3 × 5), the common prime factors are 2² and 3¹ (using the lowest power present in both), so the HCF = 2² × 3 = 4 × 3 = 12.

Meaning of the Lowest Common Multiple (LCM)

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

Finding LCM Using Prime Factorisation

To find the LCM using prime factorisation, express each number as a product of its prime factors, then multiply together all the prime factors involved, using the highest power of each prime that appears in any of the numbers. For example, to find the LCM of 36 (2² × 3²) and 60 (2² × 3 × 5), take the highest powers: 2² , 3², and 5¹, so the LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180.

Relationship Between HCF and LCM

For any two whole numbers, there is a useful relationship between their HCF and LCM: the product of the HCF and LCM of two numbers equals the product of the two numbers themselves. This can be written as: HCF × LCM = first number × second number. For example, for 12 and 18, the HCF is 6 and the LCM is 36, and indeed 6 × 36 = 216, which equals 12 × 18 = 216, confirming the relationship.

Applications of HCF

The HCF is useful in situations involving dividing quantities into equal groups or simplifying ratios. For example, if a teacher wants to divide 24 pencils and 36 erasers into identical bags with the maximum number of bags possible, each bag containing the same number of pencils and the same number of erasers, the maximum number of bags is the HCF of 24 and 36, which is 12. The HCF is also used to simplify fractions to their lowest terms by dividing both the numerator and denominator by their HCF.

Applications of LCM

The LCM is useful in situations involving events that repeat at regular intervals, or when adding and subtracting fractions with different denominators. For example, if one bus arrives every 15 minutes and another arrives every 20 minutes, both buses will arrive together again after the LCM of 15 and 20, which is 60 minutes (one hour). The LCM is also used to find the lowest common denominator when adding or subtracting fractions.

Common Mistakes with Factors and Multiples

Students often confuse factors and multiples, forgetting that factors of a number are always less than or equal to that number, while multiples of a number are always greater than or equal to that number (excluding zero). Other common mistakes include missing factor pairs when listing factors, making errors during prime factorisation, and confusing when to use HCF versus LCM in word problems — a useful tip is that HCF problems usually involve splitting or grouping into the largest possible equal parts, while LCM problems usually involve finding when repeating events coincide, or finding common denominators.

Summary

Factors are numbers that divide exactly into a given number, while multiples are the results of multiplying a number by whole numbers. The Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers, while the Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. Both can be found by listing or by using prime factorisation, and they are connected by the relationship HCF × LCM = product of the two numbers. These concepts are essential for simplifying fractions, solving grouping and sharing problems, and finding common denominators.

Practising Factors and Multiples

Building fluency with factors, multiples, HCF, and LCM takes regular practice, particularly with recognising prime numbers quickly and constructing accurate factor trees. Students are encouraged to memorise the first several prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29), practise prime factorisation with a variety of numbers, and work through word problems that require deciding whether HCF or LCM is the appropriate tool. With consistent practice, these skills become second nature and prove invaluable throughout the rest of secondary school mathematics, particularly in algebra and fraction work.

Worked Examples

Example 1: List all the factors of 42. Testing each number: 1 × 42, 2 × 21, 3 × 14, 6 × 7. Factors: 1, 2, 3, 6, 7, 14, 21, 42.

Example 2: Express 84 as a product of its prime factors. 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7. Answer: 2² × 3 × 7.

Example 3: Find the HCF of 28 and 42 using prime factorisation. 28 = 2² × 7, and 42 = 2 × 3 × 7. Common factors: 2¹ × 7¹. Answer: 14.

Example 4: Find the LCM of 8 and 12 using prime factorisation. 8 = 2³, and 12 = 2² × 3. Taking the highest powers: 2³ × 3. Answer: 24.

Example 5: Two bells ring together at 8:00 am. One bell rings every 18 minutes and the other every 24 minutes. When will they next ring together? LCM(18, 24) = 72 minutes, so they will next ring together at 9:12 am.

Student Exercise

Solve the following problems, showing all your working:

1. List all the factors of 36.
2. Express 90 as a product of its prime factors.
3. Find the HCF of 24 and 40.
4. Find the LCM of 9 and 15.
5. Find the HCF of 45 and 60 using prime factorisation.
6. Find the LCM of 12 and 18 using prime factorisation.
7. The HCF of 8 and 20 is 4. Use the relationship HCF × LCM = product of the numbers to find their LCM.
8. A trader has 48 mangoes and 60 oranges. Find the maximum number of identical baskets she can pack so that each basket has the same number of mangoes and the same number of oranges, with none left over.
9. Two lights flash together now. One flashes every 10 seconds and the other every 15 seconds. After how many seconds will they next flash together?
10. State whether 37 is a prime or a composite number, and explain why.