Section A — Chapters 1–22

Natural Numbers

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to define and identify natural numbers.
✓By the end of this lesson students will be able to perform addition, subtraction, multiplication, and division with natural numbers.
✓By the end of this lesson students will be able to identify factors, multiples, and prime numbers.
✓By the end of this lesson students will be able to understand and calculate powers (indices) and square roots of natural numbers.
✓By the end of this lesson students will be able to apply the order of operations (BODMAS) to solve problems involving natural numbers.

Key concepts

Natural Numbers

Natural numbers are the counting numbers. They are positive whole numbers starting from 1.

{1, 2, 3, 4, ...}

Addition

The process of combining two or more numbers to find their total sum. The symbol used is '+'.

Subtraction

The process of taking one number away from another to find the difference. The symbol used is '-'.

Multiplication

A quick way of adding the same number multiple times. It is also known as finding the product. The symbols used are '×' or '*'.

Division

The process of sharing a number into equal parts or finding how many times one number fits into another. The symbols used are '÷' or '/'. The result is called the quotient, and any leftover amount is the remainder.

Factors

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

Multiples

A multiple of a natural number is the result of multiplying that number by another natural number. They are like the 'times tables' for a number.

Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself.

Indices (Powers)

An index (or power) tells us how many times a base number is multiplied by itself. For example, in 2³, 2 is the base and 3 is the index. It means 2 × 2 × 2.

aⁿ = a × a × a × ... (n times)

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for square root is '√'. For example, √25 = 5 because 5 × 5 = 25.

√x = y if y × y = x

Order of Operations (BODMAS)

A set of rules that dictates the sequence in which mathematical operations should be performed to ensure a consistent result. BODMAS stands for Brackets, Orders (powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Key facts to remember

1Natural numbers are the positive whole numbers: 1, 2, 3, ...
2A factor divides a number exactly, leaving no remainder.
3A multiple is the result of multiplying a number by another natural number.
4A prime number is a natural number greater than 1 with exactly two factors: 1 and itself. (Note: 1 is not a prime number).
5Indices (powers) indicate repeated multiplication (e.g., 5³ = 5 × 5 × 5).
6The square root of a number is the value that, when multiplied by itself, gives the original number.
7Always follow the order of operations (BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Worked examples

Example 1

Calculate: (a) 345 + 187 (b) 503 - 216 (c) 48 × 15 (d) 125 ÷ 8

I(a) 345 + 187 = 532

II(b) 503 - 216 = 287

III(c) 48 × 15

IV 48 × 10 = 480

V 48 × 5 = 240

VI 480 + 240 = 720

VII(d) 125 ÷ 8

VIII 8 goes into 12 once, remainder 4.

9 Bring down 5, 8 goes into 45 five times, remainder 5.

10 So, 125 ÷ 8 = 15 remainder 5.

Answer

(a) 532 (b) 287 (c) 720 (d) 15 remainder 5

For division with natural numbers, Junior Cycle often expects remainders.

Example 2

For the number 24: (a) List all its factors. (b) List the first five multiples. (c) Is 24 a prime number? Explain why or why not.

I(a) To find factors, we look for pairs of numbers that multiply to 24:

II 1 × 24 = 24

III 2 × 12 = 24

IV 3 × 8 = 24

V 4 × 6 = 24

VI The factors are 1, 2, 3, 4, 6, 8, 12, 24.

VII(b) To find the first five multiples, multiply 24 by 1, 2, 3, 4, 5:

VIII 24 × 1 = 24

9 24 × 2 = 48

10 24 × 3 = 72

11 24 × 4 = 96

12 24 × 5 = 120

13 The first five multiples are 24, 48, 72, 96, 120.

14(c) A prime number has exactly two factors (1 and itself). 24 has more than two factors (1, 2, 3, 4, 6, 8, 12, 24). Therefore, 24 is not a prime number.

Answer

(a) 1, 2, 3, 4, 6, 8, 12, 24 (b) 24, 48, 72, 96, 120 (c) No, because it has more than two factors.

Example 3

Evaluate: (a) 3² + √49 (b) 10 × (8 - 3) ÷ 5 + 2

I(a) 3² + √49

II Calculate the power: 3² = 3 × 3 = 9

III Calculate the square root: √49 = 7 (because 7 × 7 = 49)

IV Perform addition: 9 + 7 = 16

V(b) 10 × (8 - 3) ÷ 5 + 2

VI Brackets first: (8 - 3) = 5

VII Expression becomes: 10 × 5 ÷ 5 + 2

VIII Multiplication and Division from left to right:

9 10 × 5 = 50

10 50 ÷ 5 = 10

11 Expression becomes: 10 + 2

12 Addition: 10 + 2 = 12

Answer

(a) 16 (b) 12

Remember BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction.

Common mistakes

✗Confusing factors with multiples (e.g., saying 6 is a multiple of 12 instead of a factor).
✗Incorrectly identifying 1 as a prime number.
✗Misinterpreting indices (e.g., calculating 2³ as 2 × 3 = 6 instead of 2 × 2 × 2 = 8).
✗Not following the correct order of operations, especially with division/multiplication and addition/subtraction.
✗Making calculation errors in basic arithmetic, particularly with carrying over or borrowing.

Exam tips

★Read the question carefully to understand what is being asked (e.g., 'list factors' vs. 'list multiples').
★Show all your working steps clearly, especially for multi-step problems involving the order of operations.
★Double-check your calculations, particularly for addition, subtraction, multiplication, and division.
★Memorise the definitions of key terms like 'factor', 'multiple', and 'prime number'.
★Practice applying BODMAS consistently to avoid errors in complex expressions.

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