Number operations and integers Revision Notes

What is the Highest Common Factor (HCF)?

The HCF, also known as the Greatest Common Divisor (GCD), is the largest number that divides exactly into two or more numbers without leaving a remainder. To find the HCF of two numbers:

1. List all factors of each number. For example, factors of 12 are {1, 2, 3, 4, 6, 12}. Factors of 18 are {1, 2, 3, 6, 9, 18}.
2. Identify the common factors. For 12 and 18, common factors are {1, 2, 3, 6}.
3. The HCF is the largest of these common factors. So, HCF(12, 18) = 6.

Using Prime Factorisation for HCF

For larger numbers, prime factorisation is a more efficient method. Write each number as a product of its prime factors. For example:

• 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
• 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

To find the HCF, multiply the common prime factors, raised to the lowest power they appear in either factorisation. Both 60 and 72 share 2 and 3 as prime factors.

• The lowest power of 2 is 2².
• The lowest power of 3 is 3¹.

So, HCF(60, 72) = 2² × 3 = 4 × 3 = 12.

What is the Lowest Common Multiple (LCM)?

The LCM is the smallest positive number that is a multiple of two or more numbers. To find the LCM of two numbers:

1. List the first few multiples of each number. For example, multiples of 4 are {4, 8, 12, 16, 20, 24, ...}. Multiples of 6 are {6, 12, 18, 24, 30, ...}.
2. Identify the common multiples. For 4 and 6, common multiples are {12, 24, ...}.
3. The LCM is the smallest of these common multiples. So, LCM(4, 6) = 12.

Using Prime Factorisation for LCM

Prime factorisation is also efficient for finding the LCM. Write each number as a product of its prime factors. For example:

• 60 = 2² × 3 × 5
• 72 = 2³ × 3²

To find the LCM, multiply all prime factors that appear in any of the factorisations, raised to the highest power they appear in either factorisation. The prime factors involved are 2, 3, and 5.

• The highest power of 2 is 2³.
• The highest power of 3 is 3².
• The highest power of 5 is 5¹.

So, LCM(60, 72) = 2³ × 3² × 5 = 8 × 9 × 5 = 72 × 5 = 360. Remember the relationship: HCF(a, b) × LCM(a, b) = a × b. For 60 and 72: 12 × 360 = 4320, and 60 × 72 = 4320. This confirms the values.

What is the Highest Common Factor (HCF)?

The HCF of two or more numbers is the largest number that divides exactly into all of them without leaving a remainder. To find the HCF, you can list all the factors of each number and identify the largest factor that they share. For instance, factors of 20 are 1, 2, 4, 5, 10, 20, and factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The common factors are 1, 2, 5, 10. The highest of these is 10, so HCF(20, 30) = 10.

What is the Lowest Common Multiple (LCM)?

The LCM of two or more numbers is the smallest positive number that is a multiple of all of them. To find the LCM, you can list the multiples of each number until you find the first multiple they share. For example, multiples of 6 are 6, 12, 18, 24, 30, ... and multiples of 8 are 8, 16, 24, 32, .... The first common multiple is 24, so LCM(6, 8) = 24. This method works well for smaller numbers.

Using Prime Factorisation for HCF

For larger numbers, prime factorisation is a more efficient method. First, write each number as a product of its prime factors. To find the HCF, identify all common prime factors and multiply them, using the lowest power (exponent) that each common prime factor appears in *any* of the numbers' factorisations. For example, if 72 = 2³ × 3² and 90 = 2 × 3² × 5. The common prime factors are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3². So, HCF(72, 90) = 2¹ × 3² = 2 × 9 = 18.

Using Prime Factorisation for LCM

To find the LCM using prime factorisation, again write each number as a product of its prime factors. For the LCM, identify ALL prime factors that appear in *any* of the numbers' factorisations (both common and uncommon). Multiply these prime factors, using the highest power (exponent) that each prime factor appears in *any* of the numbers' factorisations. Using the previous example: 72 = 2³ × 3² and 90 = 2 × 3² × 5. The prime factors are 2, 3, and 5. The highest power of 2 is 2³, the highest power of 3 is 3², and the highest power of 5 is 5¹. So, LCM(72, 90) = 2³ × 3² × 5¹ = 8 × 9 × 5 = 360.

What is the Priority of Operations?

The priority of operations is a set of rules that dictate the order in which mathematical operations should be performed in a single expression. This standard order ensures that everyone who evaluates an expression will arrive at the same answer. It's often remembered using acronyms like BODMAS or BIDMAS. Without these rules, calculations could lead to multiple different results, making mathematics inconsistent.

BODMAS / BIDMAS Explained

BODMAS stands for Brackets, Orders (powers/indices), Division, Multiplication, Addition, Subtraction. BIDMAS uses 'Indices' instead of 'Orders'. The order is: 1.

B / P - Brackets / Parentheses (deal with anything inside these first). 2.

O / I - Orders / Indices (powers like x², cube roots, etc.). 3.

DM - Division and Multiplication (these have equal priority). Work from left to right as you encounter them. 4.

AS - Addition and Subtraction (these also have equal priority). Work from left to right as you encounter them.

Applying the Rules: Left to Right Principle

It's a common mistake to think that division always comes before multiplication, or addition before subtraction. This is incorrect. Division and Multiplication share the same level of priority. You perform whichever comes first from left to right in the expression. The same goes for Addition and Subtraction. For example, in 10 - 3 + 2, you subtract 3 from 10 first, then add 2 (result 9), not add 3 + 2 then subtract from 10 (result 5).

What are Inverse Operations?

Inverse operations are mathematical operations that reverse the effect of each other. They allow you to get back to the starting value after performing an operation. Think of them as 'un-doing' an action. For example, if you add something to a number, to get back to the original number, you would subtract that same amount. This concept is essential for solving equations and checking your work in various mathematical contexts. Always remember that for every operation, there is an inverse operation.

Common Pairs of Inverse Operations

There are key pairs of inverse operations you need to know:

1. Addition and Subtraction: If you add a number, you can subtract the same number to reverse the process. For example, starting with 7, adding 3 gives 10; subtracting 3 from 10 gives back 7.
2. Multiplication and Division: If you multiply by a number, you can divide by the same number to reverse the process. For example, starting with 5, multiplying by 2 gives 10; dividing 10 by 2 gives back 5.

Understanding these pairs is crucial for solving algebraic equations and verifying calculations effectively.

Using Inverse Operations to Solve Problems

Inverse operations are primarily used to solve for unknown values in equations. The goal is to isolate the variable by 'undoing' the operations performed on it. For instance, to solve 'x + 5 = 12', you'd use the inverse of addition (subtraction) by subtracting 5 from both sides: 'x = 12 - 5', so 'x = 7'. Similarly, to solve '4x = 20', you'd use the inverse of multiplication (division) by dividing both sides by 4: 'x = 20 / 4', so 'x = 5'. Always apply the inverse operation to both sides of an equation to maintain balance.

Checking Your Work with Inverse Operations

A great way to check if your answer to a calculation or an equation is correct is to use inverse operations. If you solved 'x - 3 = 8' and found 'x = 11', you can check this by substituting '11' back into the original equation: '11 - 3 = 8'. Since this is true, your answer is correct. Or, if you performed a division like '15 / 3 = 5', you can check it by performing the inverse multiplication: '5 * 3 = 15'. This method provides a reliable way to verify the accuracy of your mathematical work.