Number
Structure and Calculation: Four Operations, Factors, Primes, HCF/LCM, Powers, Roots, and Order of Operations
Year 10 · Year 11
- ✓By the end of this lesson students will be able to perform calculations involving the four basic operations with integers and decimals.
- ✓By the end of this lesson students will be able to identify factors, multiples, and prime numbers, and use prime factorisation to find the HCF and LCM of two or more numbers.
- ✓By the end of this lesson students will be able to calculate powers (indices) and roots (square roots, cube roots) of numbers.
- ✓By the end of this lesson students will be able to apply the order of operations (BIDMAS/BODMAS) correctly to evaluate complex expressions.
- ✓By the end of this lesson students will be able to solve problems that combine these number concepts.
Key concepts
The four basic arithmetic operations are addition (+), subtraction (-), multiplication (×), and division (÷). These are fundamental to all numerical calculations.
A factor of a whole number is another whole number that divides exactly into it, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
A multiple of a number is the result of multiplying that number by an integer. Multiples are essentially numbers in the number's times table. For example, multiples of 5 are 5, 10, 15, 20, ...
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, ...
The Highest Common Factor (HCF) of two or more numbers is the largest number that divides exactly into all of them. It can be found by identifying common prime factors and multiplying them.
The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all of them. It can be found using prime factorisation by taking the highest power of all prime factors present in any of the numbers.
A power (or index) indicates how many times a base number is multiplied by itself. For example, in aⁿ, 'a' is the base and 'n' is the power/index. a² means a × a, a³ means a × a × a, etc.
A root is the inverse operation of a power. The square root (√) of a number 'x' is a number 'y' such that y² = x. The cube root (³√) of 'x' is 'y' such that y³ = x. Higher roots also exist.
BIDMAS (or BODMAS) is an acronym used to remember the correct order of operations in mathematical expressions: Brackets, Indices (or Orders), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
Key facts to remember
- 1A prime number has exactly two factors: 1 and itself. The number 1 is not a prime number.
- 2The number 2 is the only even prime number.
- 3Prime factorisation is the process of writing a number as a product of its prime factors.
- 4BIDMAS/BODMAS order: Brackets, Indices (Powers/Roots), Division/Multiplication (left to right), Addition/Subtraction (left to right).
- 5Any non-zero number raised to the power of 0 is 1 (e.g., 5⁰ = 1).
- 6Any number raised to the power of 1 is the number itself (e.g., 7¹ = 7).
- 7HCF can be found by multiplying the common prime factors raised to their lowest powers.
- 8LCM can be found by multiplying all prime factors (common and uncommon) raised to their highest powers.
Worked examples
Example 1
Evaluate: 18 + 6 × (15 - 7) ÷ 2²
Answer
30
Remember to strictly follow the BIDMAS order, especially for division and multiplication, and addition and subtraction, which are performed from left to right.
Example 2
Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 60 and 96.
Answer
HCF = 12, LCM = 480
Drawing a Venn diagram can be a helpful visual aid for finding HCF and LCM from prime factors.
Example 3
Evaluate: (³√125 + 4²) × (20 - √81)
Answer
231
Break down complex expressions into smaller, manageable parts, working from the innermost brackets outwards.
Common mistakes
- ✗Incorrectly applying the order of operations, especially performing addition/subtraction before multiplication/division.
- ✗Confusing the HCF and LCM, or incorrectly using prime factorisation to find them.
- ✗Miscalculating powers (e.g., 3² = 6 instead of 9) or roots.
- ✗Treating division and multiplication (or addition and subtraction) as having a strict order, rather than equal priority and working from left to right.
- ✗Forgetting that 1 is not a prime number.
Exam tips
- ★Always show all your working steps clearly, even for calculator questions, as method marks are often awarded.
- ★Write down 'BIDMAS' or 'BODMAS' at the top of your page for complex calculations to remind yourself of the correct order.
- ★Use prime factorisation for finding HCF and LCM, especially for larger numbers, to avoid errors.
- ★Double-check your calculations, particularly when dealing with negative numbers or multiple operations.
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