Number Foundations: Operations, Factors, and Primes

TL;DR

This topic is all about building strong number sense by mastering the basic operations, understanding how numbers are built from their factors, and identifying the special numbers called primes. You'll learn how to break numbers down and put them back together using multiplication and division. This foundation is crucial for tackling more complex algebra and number theory in the future.

1. The Mental Model

Think of numbers as LEGO bricks. Some are basic single bricks (primes), while others are creations built from these smaller bricks (composite numbers). Understanding how to add, subtract, multiply, and divide these bricks, and how to find the individual bricks that make up a bigger structure, is what number foundations is all about.

2. The Core Material

You already know the four basic operations: addition, subtraction, multiplication, and division. We'll quickly review some key ideas about them and then dive into factors and primes.

Order of Operations (BIDMAS/BODMAS)

Remember BIDMAS (or BODMAS)! It's the golden rule for solving sums with multiple operations:

• Brackets first
• Indices (powers/roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

For example, $2 + 3 \times 4$ is not $(2+3) \times 4 = 20$. It's $2 + (3 \times 4) = 2 + 12 = 14$.

Factors

A factor of a number is any whole number that divides into it exactly, with no remainder.

For example, for the number 12:
* $1 \times 12 = 12$
* $2 \times 6 = 12$
* $3 \times 4 = 12$

So, the factors of 12 are 1, 2, 3, 4, 6, and 12. Notice that factors always come in pairs (unless the number is a perfect square, like 9, where $3 \times 3 = 9$).

Multiples

A multiple of a number is what you get when you multiply that number by another whole number.

For example, the multiples of 3 are 3, 6, 9, 12, 15, ... ($3 \times 1, 3 \times 2, 3 \times 3$, etc.)

Prime Numbers

A prime number is a whole number greater than 1 that has only two factors: 1 and itself.

Examples:
* 2 (factors: 1, 2) - it's the only even prime number!
* 3 (factors: 1, 3)
* 5 (factors: 1, 5)
* 7 (factors: 1, 7)
* 11 (factors: 1, 11)

Numbers greater than 1 that are not prime are called composite numbers. They have more than two factors. For example, 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 9 (factors: 1, 3, 9).

Important: The number 1 is neither prime nor composite. It only has one factor (itself).

Prime Factorisation

Prime factorisation is when you break down a composite number into a product of its prime factors. Every composite number has a unique prime factorisation (this is called the Fundamental Theorem of Arithmetic – sounds fancy, but it just means it's a unique recipe!).

You can use a factor tree to do this:

graph TD
    A[Start with your number: 60] --> B(Break into any two factors: 60 = 6 x 10)
    B --> C(Is 6 prime? No.)
    B --> D(Is 10 prime? No.)
    C --> E(Break 6 into factors: 6 = 2 x 3)
    E --> F(Is 2 prime? Yes.)
    E --> G(Is 3 prime? Yes.)
    D --> H(Break 10 into factors: 10 = 2 x 5)
    H --> I(Is 2 prime? Yes.)
    H --> J(Is 5 prime? Yes.)
    K{Collect all the prime factors}
    F --> K
    G --> K
    I --> K
    J --> K
    K --> L["Result: 60 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5"]

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

These are super useful for fractions and problem-solving.

• HCF: The largest factor that two or more numbers share.
  To find the HCF of 12 and 18:
  Factors of 12: 1, 2, 3, 4, 6, 12
  Factors of 18: 1, 2, 3, 6, 9, 18
  Common factors are 1, 2, 3, 6. The HCF is 6.

• LCM: The smallest positive number that is a multiple of two or more numbers.
  To find the LCM of 4 and 6:
  Multiples of 4: 4, 8, 12, 16, 20, 24, ...
  Multiples of 6: 6, 12, 18, 24, 30, ...
  Common multiples are 12, 24, ... The LCM is 12.

Using prime factorisation can make finding HCF and LCM for bigger numbers much easier.

3. Worked Example

Let's find the HCF and LCM of 36 and 90 using prime factorisation.

1. Prime factorise 36:
   $36 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

2. Prime factorise 90:
   $90 = 2 \times 45 = 2 \times 3 \times 15 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5$

3. Find the HCF:
   Look at the prime factors common to both numbers, and take the lowest power for each common prime.
   Both have a '2' and a '3'.
   Lowest power of 2 is $2^1$ (from 90's factorisation).
   Lowest power of 3 is $3^2$ (both have $3^2$).
   HCF = $2^1 \times 3^2 = 2 \times 9 = 18$.

4. Find the LCM:
   Look at all the prime factors from both numbers, and take the highest power for each prime.
   Primes involved: 2, 3, 5.
   Highest power of 2 is $2^2$ (from 36's factorisation).
   Highest power of 3 is $3^2$ (from both).
   Highest power of 5 is $5^1$ (from 90's factorisation).
   LCM = $2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 36 \times 5 = 180$.

4. Key Takeaways

• BIDMAS/BODMAS ensures you always get the correct answer when multiple operations are involved.
• Factors divide a number exactly, while multiples are found by multiplying a number by other whole numbers.
• Prime numbers are the building blocks, having only two factors: 1 and themselves.
• Prime factorisation breaks down any composite number into a unique product of prime numbers.
• HCF is the biggest number that divides into all given numbers exactly.
• LCM is the smallest number that all given numbers divide into exactly.
• Understanding factors and primes makes HCF and LCM calculations easier, especially for larger numbers.

Common Mistakes to Avoid:
- Forgetting BIDMAS/BODMAS and doing operations in the wrong order.
- Including 1 as a prime number (it's not!).
- Confusing factors and multiples – factors are smaller or equal, multiples are larger or equal.
- Not finding all prime factors when doing prime factorisation.
- When finding HCF/LCM using prime factorisation, mixing up 'lowest power' for HCF and 'highest power' for LCM.

5. Now Try It

Spend 15 minutes working through this challenge:

1. Calculate: $25 - (3 \times 4) + 6^2 \div 2$.
2. List all the factors of 48.
3. Write down the first five prime numbers.
4. Find the prime factorisation of 72.
5. Using your prime factorisations, find the HCF and LCM of 48 and 72.

Success looks like getting the correct answers for all parts, especially for the HCF and LCM, by showing your working clearly.