Multiples and Factors
Understanding multiples and factors is essential for solving various problems in GCSE Maths. These concepts are the building blocks for more complex topics like prime factorisation, Highest Common Factor (HCF), and Lowest Common Multiple (LCM). This guide will help you understand these topics and give you a clear understanding of how to work with them.
What are Multiples and Factors?
Multiples
A multiple of a number is the result of multiplying that number by any integer. For example, the multiples of 4 are:
$$4,8,12,16,20,…$$
In general, the multiples of a number $n$ are written as $n,2n,3n,4n,…$
Factors
A factor of a number is a whole number that divides exactly into that number, leaving no remainder. For example, the factors of 12 are:
$$1,2,3,4,6,12$$
In other words, factors are numbers that divide evenly into a given number.
Powers and Roots
Powers
A power of a number is the result of multiplying that number by itself a certain number of times. For example:
$$2^3 = 2 \times 2 \times 2 = 8$$
The number $2$ is the base, and the exponent $3$ indicates how many times the base is multiplied by itself.
Square Roots and Cube Roots
The square root of a number is the value that, when multiplied by itself, gives the original number. For example:
$$16\sqrt{16} = 4 \quad \text{because} \quad 4 \times 4 = 16$$
The cube root of a number is the value that, when multiplied by itself three times, gives the original number. For example:
$$\sqrt[3]{27} = 3 \quad \text{because} \quad 3 \times 3 \times 3 = 27$$
Calculating Square and Cube Roots Using the Product of Primes
To calculate square and cube roots, it’s often helpful to prime factorise the number. Prime factorisation involves breaking down a number into its prime factors (numbers that can only be divided by 1 and themselves).
Example 1: Square Root of 36
First, find the prime factorisation of 36:
$$36 = 2 \times 2 \times 3 \times 3$$
Group the prime factors in pairs:
$$36 = (2 \times 3) \times (2 \times 3)$$
Take one factor from each pair:
$$\sqrt{36} = 2 \times 3 = 6$$
Highest Common Factor (HCF) and Lowest Common Multiple (LCM)
The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) are two important concepts in dealing with numbers.
Highest Common Factor (HCF)
The HCF of two or more numbers is the largest number that divides exactly into all of them. To find the HCF, list all the factors of each number and choose the largest common factor.
Example: Find the HCF of 24 and 30
- List the factors of each number:
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- The common factors are: 1, 2, 3, 6
- The HCF is the largest common factor: 6
Lowest Common Multiple (LCM)
The LCM of two or more numbers is the smallest number that is a multiple of all of them. To find the LCM, list the multiples of each number and choose the smallest common multiple.
Example: Find the LCM of 4 and 5
- List the multiples of each number:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …
- The smallest common multiple is 20, so the LCM of 4 and 5 is 20.
HCF and LCM Using Prime Factors
A more systematic approach to finding the HCF and LCM involves using prime factorisation.
Finding the HCF Using Prime Factors
- Prime factorise each number.
- Identify the common prime factors.
- Choose the lowest powers of the common prime factors.
- Multiply the common factors to find the HCF.
Example: Find the HCF of 36 and 60 using prime factors
Prime factorisation of 36:
$$36 = 2^2 \times 3^2$$
Prime factorisation of 60:
$$60 = 2^2 \times 3 \times 5$$
The common prime factors are $2^2$ and $3$.
Multiply the lowest powers of the common factors:
$$\text{HCF} = 2^2 \times 3 = 4 \times 3 = 12$$
Finding the LCM Using Prime Factors
- Prime factorise each number.
- Identify all the prime factors.
- For each prime factor, choose the highest power.
- Multiply the highest powers of all the prime factors to find the LCM.
Example: Find the LCM of 36 and 60 using prime factors
Prime factorisation of 36:
$$36 = 2^2 \times 3^2$$
Prime factorisation of 60:
$$60 = 2^2 \times 3 \times 5$$
The prime factors are $2^2$, $3^2$, and $5$.
Multiply the highest powers of each prime factor:
$$\text{LCM} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$$
Summary
- Multiples are numbers you get by multiplying a given number by integers.
- Factors are numbers that divide exactly into a given number.
- Powers and roots involve raising numbers to exponents or finding the value that, when raised to a certain power, gives the original number.
- HCF is the largest factor common to two or more numbers, and LCM is the smallest multiple common to two or more numbers.
- Prime factorisation can help you find both HCF and LCM more easily by identifying the common or highest powers of prime factors.
By Understanding these concepts, you’ll be better equipped to solve problems involving multiples, factors, and prime factorisation in your GCSE exams.
