Whole number theory

Welcome to Whole Number Theory!

Hi there! Welcome to your study notes for Whole Number Theory. This chapter is part of the "Number Operations and Integers" section of your OCR GCSE (9-1) Mathematics course. Think of whole numbers as the "building blocks" of everything else you'll do in math. Once you master these terms and techniques, you'll have a much easier time with fractions, algebra, and beyond!

In this guide, we will break down the language of numbers so that it makes perfect sense. Don't worry if some of this feels like a lot to remember—we’ll use plenty of tricks to help it stick!

1. The Basics: Definitions and Terms

Before we dive into calculations, we need to speak the same language. Here are the most important terms you need to know:

Odd and Even Numbers

Even numbers are integers that can be divided by 2 with no remainder. They always end in 0, 2, 4, 6, or 8.
Example: 24, 150, 3,008.

Odd numbers are integers that cannot be divided exactly by 2. They always end in 1, 3, 5, 7, or 9.
Example: 13, 57, 1,001.

Place Value

Place value tells us how much a digit is worth based on its position in a number.
Analogy: Think of a number like a set of containers. The position of the digit tells you if you have "Single Items," "Boxes of 10," or "Crates of 100."

In the number 5,238:
The 5 is in the thousands place (\(5,000\)).
The 2 is in the hundreds place (\(200\)).
The 3 is in the tens place (\(30\)).
The 8 is in the units (or ones) place (\(8\)).

Square, Cube, and Roots

Square Numbers: The result of multiplying a whole number by itself.
\(3 \times 3 = 9\) (So, 9 is a square number).
Cube Numbers: The result of multiplying a number by itself, then by itself again.
\(2 \times 2 \times 2 = 8\) (So, 8 is a cube number).

Roots: The "opposite" of squaring or cubing.
The square root (\(\sqrt{}\)) of 25 is 5 because \(5 \times 5 = 25\).
The cube root (\(\sqrt[3]{}\)) of 27 is 3 because \(3 \times 3 \times 3 = 27\).

Key Takeaway: Knowing your definitions is half the battle! Even and odd are about divisibility by 2, while place value determines the "weight" of a digit.

2. Factors and Multiples

Struggling with these? You aren't alone! Here is the easiest way to tell them apart:

Factors (Divisors)

Factors are the numbers that fit exactly into another number. They are always smaller than or equal to the number you are looking at.
Memory Aid: Factors are Few (there is a limited amount of them).

How to find factors of 12:
Always work in pairs to make sure you don't miss any!
1 and 12 (\(1 \times 12 = 12\))
2 and 6 (\(2 \times 6 = 12\))
3 and 4 (\(3 \times 4 = 12\))
The factors of 12 are: 1, 2, 3, 4, 6, 12.

Multiples

Multiples are what you get when you multiply a number by other whole numbers (like a "times table"). These go on forever!
Memory Aid: Multiples Multiply (they get bigger).

The multiples of 5 are: 5, 10, 15, 20, 25... and so on.

Common Mistake: Many students swap these. Just remember: Factors "go into" a number; Multiples "grow out" of a number.

Quick Review:
- Factors of 10: 1, 2, 5, 10.
- Multiples of 10: 10, 20, 30, 40...

3. Prime Numbers

A Prime Number is a whole number greater than 1 that has exactly two factors: 1 and itself.

Identifying Primes

You should memorize the primes less than 20:
2, 3, 5, 7, 11, 13, 17, 19

Did you know?
The number 2 is the only even prime number! All other even numbers can be divided by 2, so they aren't prime. Also, 1 is NOT a prime number because it only has one factor (itself), and primes must have exactly two.

Product of Prime Factors

Every whole number (except 1) can be broken down into a unique "recipe" of prime numbers. We often use a Factor Tree to find this.

Step-by-step: Prime Factors of 24
1. Start with 24. Split it into any two factors: \(2 \times 12\).
2. 2 is prime, so circle it!
3. Split 12: \(2 \times 6\). Circle the 2.
4. Split 6: \(2 \times 3\). Both 2 and 3 are prime, so circle them.
5. Write them down: \(24 = 2 \times 2 \times 2 \times 3\).
6. Power Notation: Instead of writing \(2 \times 2 \times 2\), we can write \(2^3\).
So, \(24 = 2^3 \times 3\).

Key Takeaway: Prime factors are like the "DNA" of a number. No matter how you start your factor tree, you will always end up with the same prime numbers at the bottom!

4. HCF and LCM

These terms often appear in exam questions involving "finding the smallest amount" or "sharing things into groups."

Highest Common Factor (HCF)

The HCF is the biggest number that is a factor of two (or more) numbers.
Example: 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
The biggest number in both lists is 6.

Lowest Common Multiple (LCM)

The LCM is the smallest number that appears in the multiples list of both numbers.
Example: Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 6: 6, 12, 18, 24
The first (smallest) number to appear in both lists is 12.

Using Prime Factorization for HCF/LCM

If the numbers are large, listing takes too long. Use your prime factors instead!
1. Find the prime factors of both numbers.
2. To find the HCF: Multiply the prime factors that both numbers share.
3. To find the LCM: Multiply every prime factor that appears, but only the highest power of each.

Key Takeaway: Highest Common Factor focuses on what is shared. Lowest Common Multiple is the first place two different "times tables" meet.

Final Checklist for Success

Before you move on, make sure you can answer these:
- Can I identify if a number is odd or even?
- Do I know my prime numbers up to 20?
- Can I draw a factor tree to find prime factors?
- Can I find the HCF and LCM of two numbers?
- Do I know that 1 is not prime and 2 is the only even prime?

You've got this! Whole number theory is the foundation of your math journey. Keep practicing those factor trees and you'll be an expert in no time!