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Natural Numbers & Integers
ℕ
Introduction to Natural Numbers
Counting • Order

In this course, the natural numbers are 1, 2, 3, 4, 5, …. We write this as ℕ = {1, 2, 3, 4, …}.

We use natural numbers to count and to show position in an order or list.

  • 1 Counting: 1 book, 2 books, 3 books, …
  • 2 Order: 1st place, 2nd place, 3rd place, …

Natural Numbers & Place Value

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Prime and Composite Numbers
Divisors • Factors

A prime number has exactly two factors — 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17.

A composite number has more than two factors.
Examples: 4, 6, 8, 9, 10, 12.

The number 1 is neither prime nor composite.

  • ✓ 2 is the only even prime.
  • ✓ Every whole number greater than 1 is prime or composite.
  • ✓ Prime numbers are the building blocks of all integers.

Find the Prime Factors

Find the prime factors of:
(i) 28  (ii) 60
(i) 28
Draw a factor tree:
28
├── 2 × 14
│  └── 2 × 7
Prime factors: \(2 \times 2 \times 7 = 2^{2}\times7\)
(ii) 60
Draw a factor tree:
60
├── 2 × 30
│  ├── 2 × 15
│  │  └── 3 × 5
Prime factors: \(2 \times 2 \times 3 \times 5 = 2^{2}\times3\times5\)
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Highest Common Factor (HCF)
Factors • Common Factors • HCF

A factor of a number divides it exactly, with no remainder.

A common factor is a number that divides two or more numbers exactly.

The Highest Common Factor (HCF) is the largest of these shared factors.

  • 1 Factors of 12: 1, 2, 3, 4, 6, 12
  • 2 Factors of 18: 1, 2, 3, 6, 9, 18
  • 3 Common factors: 1, 2, 3, 6
  • 4 HCF(12, 18) = 6

Find the Highest Common Factor

Find the highest common factor of \(36\) and \(54\).
Method 1: Using prime factors
Write each number as a product of prime factors.
\(36 = 2 \times 2 \times 3 \times 3 = 2^{2}\times3^{2}\)
\(54 = 2 \times 3 \times 3 \times 3 = 2\times3^{3}\)
Common primes: one factor \(2\) and two factors \(3\).
\(\text{HCF} = 2 \times 3^{2} = 2 \times 9 = 18\).
Highest common factor: \(18\).
Method 2: Listing factors
List the factors of each number.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
The common factors are \(1, 2, 3, 6, 9, 18\). The highest of these is \(18\).
So the HCF of 36 and 54 is \(18\).
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HCF Using Prime Factors

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Multiples and LCM
Multiples • Common Multiples • LCM

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

Multiples of 3: 3, 6, 9, 12, 15, 18, …
Multiples of 4: 4, 8, 12, 16, 20, …

A common multiple of two numbers is a number that is a multiple of both.

The Lowest Common Multiple (LCM) of two numbers is the smallest common multiple.

  • 1 12 is a common multiple of 3 and 4.
  • 2 The LCM of 3 and 4 is 12.
  • 3 LCM is used to line up repeats, e.g. bus timetables or patterns.

Find the Lowest Common Multiple

Find the lowest common multiple (LCM) of \(8\), \(12\) and \(18\).
Method 1: Using prime factors
Write each number as a product of prime factors.
\(8 = 2 \times 2 \times 2 = 2^{3}\)
\(12 = 2 \times 2 \times 3 = 2^{2}\times3\)
\(18 = 2 \times 3 \times 3 = 2\times3^{2}\)
For the LCM, take the highest power of each prime: \(2^{3}\) and \(3^{2}\).
\(\text{LCM} = 2^{3}\times3^{2} = 8\times9 = 72\).
So the lowest common multiple is \(72\).
Method 2: Listing multiples (quick check)
List some multiples of each number.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, …
Multiples of 12: 12, 24, 36, 48, 60, 72, …
Multiples of 18: 18, 36, 54, 72, …
The first multiple they all have in common is \(72\).
Hence, \(\text{LCM}(8,12,18)=72\).
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LCM Using Prime Factors

HCF & LCM Quiz

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Introduction to Integers
Positive • Negative • Zero • Operations

The set of integers includes all positive and negative whole numbers and zero: ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}

Integers are used for values above and below zero — temperature, altitude, or bank balance.

  • 1 Positive: +4, +7, +12
  • 2 Negative: −3, −8, −15
  • 3 Zero: neither positive nor negative

Examples of Operations with Integers

(+3) + (+4) = +7Adding two positives
(+5) + (−8) = −3Positive plus a larger negative
(−6) + (−1) = −7Adding two negatives
5 − (−3) = +8Subtracting a negative
(−4) − (+2) = −6Negative minus positive
(−7) − (−5) = −2Negative minus negative
(+3) × (−2) = −6Positive × negative
(−4) × (−3) = +12Negative × negative
(−5) × 2 = −10Negative × positive
(+12) ÷ (−3) = −4Positive ÷ negative
(−15) ÷ (−3) = +5Negative ÷ negative
(−8) ÷ 4 = −2Negative ÷ positive

Operations with Integers Quiz

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Order of Operations
Brackets • Indices • Roots • Division • Multiplication • Addition • Subtraction

When a calculation has several operations, we use BIRDMAS to decide the order in which to work:

  • BBrackets
  • IIndices
  • RRoots
  • DDivision
  • MMultiplication
  • AAddition
  • SSubtraction

We work from left to right at each stage, following this order.

Quick comparisons
3 + 4 × 2 = 11 Multiply first: 4 × 2 = 8, then 3 + 8
(3 + 4) × 2 = 14 Brackets first: 3 + 4 = 7, then 7 × 2
2 × 5² = 50 Index first: 5² = 25, then 2 × 25
√16 + 3² = 13 Root and index first: 4 + 9

Order of Operations (BIMDAS)

Work out:
(i) \(12 + 8 \div 2 + 4 \times 3\)
(ii) \(7 \times 8 - 3 - (72 \div 8)\)
(i) \(12 + 8 \div 2 + 4 \times 3\)
Step 1 — Division: \(8 \div 2 = 4\)
Now \(12 + 4 + 4 \times 3\)
Step 2 — Multiplication: \(4 \times 3 = 12\)
Now \(12 + 4 + 12\)
Step 3 — Addition: \(12 + 4 + 12 = 28\)
Answer: 28
(ii) \(7 \times 8 - 3 - (72 \div 8)\)
Step 1 — Brackets: \(72 \div 8 = 9\)
Now \(7 \times 8 - 3 - 9\)
Step 2 — Multiplication: \(7 \times 8 = 56\)
Now \(56 - 3 - 9\)
Step 3 — Subtraction: \(56 - 3 - 9 = 44\)
Answer: 44
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Associative and Commutative Properties of Numbers

Order of Operations (BIRDMAS) Quiz

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Use BIRDMAS: Brackets, Indices/Roots, Division & Multiplication, Addition & Subtraction. © onlinemaths.org