Natural numbers
The counting numbers
\(1,2,3,4,\ldots\)
In this course, natural numbers start at 1.
Chapter 1
Number
Build the number skills that support almost every later topic in Mathematics. These notes focus on recognising number types, using prime factors, and choosing HCF or LCM efficiently in both numerical and practical problems.
By the end of this chapter, you should be able to
- identify natural numbers, integers, prime numbers, square numbers, cube numbers, rational numbers and irrational numbers;
- find reciprocals of non-zero numbers;
- write whole numbers as products of prime factors using index notation;
- use prime factors to reason about square and cube numbers;
- find the highest common factor (HCF) and lowest common multiple (LCM);
- recognise whether a real-life problem needs an HCF or an LCM method.
1
Subtopic 1
Types of number
Different types of number overlap. For example, the number \(9\) is a natural number, an integer and a rational number. In an exam, classify a number using the definition that the question asks for rather than assuming each number belongs to only one group.
Exam distinction: an integer, a terminating decimal and a recurring decimal are all rational. A square root is rational when it simplifies to a rational value, such as \(\sqrt{49}=7\), but \(\sqrt{7}\) is irrational. The reciprocal of zero is undefined.
Integers
Positive and negative whole numbers, including zero.
\(\ldots,-2,-1,0,1,2,\ldots\)
Rational numbers
Numbers that can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\neq0\).
Integers, terminating decimals and recurring decimals are rational.
Irrational numbers
Numbers that cannot be written as an exact fraction of two integers.
Their decimal forms do not terminate and do not recur, e.g. \(\sqrt2\) and \(\pi\).
Prime numbers
Whole numbers with exactly two positive factors: 1 and themselves.
For example, \(2,3,5,7,11\). The number 1 is not prime.
Square and cube numbers
A square number has the form \(n^2\); a cube number has the form \(n^3\).
Examples: \(25=5^2\), \(64=4^3=8^2\).
Rational or irrational?
| Form | Type | Reason |
|---|---|---|
| \(-12\) | Rational | Every integer can be written as a fraction, e.g. \(-12=\frac{-12}{1}\). |
| \(0.375\) | Rational | It terminates: \(0.375=\frac{375}{1000}=\frac38\). |
| \(0.272727\ldots\) | Rational | The decimal repeats. |
| \(\sqrt{49}\) | Rational | \(\sqrt{49}=7\). |
| \(\sqrt{10}\) | Irrational | It is not the square root of a perfect square and its decimal neither terminates nor recurs. |
| \(3\pi\) | Irrational | A non-zero rational multiple of \(\pi\) remains irrational. |
Reciprocals
For a non-zero number \(x\), its reciprocal is \(\frac1x\). Multiplying a number by its reciprocal gives 1.
\[5\longrightarrow \frac15\]
\[\frac37\longrightarrow \frac73\]
\[-4\longrightarrow -\frac14\]
Worked example 1 • Easy
Classifying numbers
Classify each number as integer, rational or irrational: \(-6,\ 0.45,\ \sqrt{36},\ \sqrt7\).
Step 1: \(-6\) is a whole number, so it is an integer. It is also rational because \(-6=\frac{-6}{1}\).
Step 2: \(0.45\) terminates, so it is rational.
Step 3: \(\sqrt{36}=6\), so it is an integer and rational.
Step 4: \(\sqrt7\) is not the square root of a perfect square, so it is irrational.
Worked example 2 • Medium
Justifying a rational number
Explain why \(1.625\) is rational.
A terminating decimal can be written as a fraction:
\[1.625=\frac{1625}{1000}=\frac{13}{8}.\]Because \(13\) and \(8\) are integers and the denominator is not zero, \(1.625\) is rational.
Worked example 3 • Exam-style
Choosing an irrational number in an interval
Give an irrational number between 5 and 6, and justify your answer.
Since \(5^2=25\) and \(6^2=36\), choose a non-square integer between 25 and 36, for example 30.
\[25<30<36\quad\Rightarrow\quad 5<\sqrt{30}<6.\]Because 30 is not a perfect square, \(\sqrt{30}\) is irrational. So \(\boxed{\sqrt{30}}\) is a valid answer.
Common exam traps
- Do not call every square root irrational: \(\sqrt{64}=8\) is rational.
- Do not call 1 a prime number. It has only one positive factor.
- A recurring decimal is rational even when it has infinitely many digits.
- Remember that a number can belong to more than one category.
Additional worked example 1 · Medium
Sort numbers with reasons
Classify \(-\frac{11}{4}\), \(\sqrt{81}\), and \(2\pi\) as rational or irrational.
Step 1: \(-\frac{11}{4}\) is a ratio of integers, so it is rational.
Step 2: \(\sqrt{81}=9\) is an integer, so it is rational.
Step 3: \(2\pi\) is irrational because a non-zero rational multiple of \(\pi\) is irrational.
Additional worked example 2 · Exam-Style
Reciprocal and classification
A number is \(-1.25\). Find its reciprocal as a fraction and explain its number type.
Step 1: Write \(-1.25=-\frac54\).
Step 2: Invert numerator and denominator: \(\frac{1}{-5/4}=-\frac45\).
Step 3: The reciprocal is rational because it is a fraction of two integers.
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
1.1
Easy [1 mark]
Write the reciprocal of \(\frac{7}{9}\).
Show Answer
Step 1: Exchange numerator and denominator.
Step 2: \(\boxed{\frac97}\).
1.2
Easy [1 mark]
Is \(\sqrt{121}\) rational or irrational?
Show Answer
Step 1: \(\sqrt{121}=11\).
Step 2: 11 is an integer, hence rational.
1.3
Medium [2 marks]
Write \(0.375\) as a fraction and classify it.
Show Answer
Step 1: \(0.375=\frac{375}{1000}\).
Step 2: Divide numerator and denominator by 125: \(\boxed{\frac38}\), rational.
1.4
Exam-Style [3–4 marks]
Identify which of \(\sqrt{18}\), \(\sqrt{\frac{9}{16}}\) and \(0.\overline{27}\) are rational.
Show Answer
Step 1: \(\sqrt{18}=3\sqrt2\) is irrational.
Step 2: \(\sqrt{\frac9{16}}=\frac34\) is rational.
Step 3: \(0.\overline{27}=\frac{27}{99}=\frac3{11}\) is rational.
2
Subtopic 2
Prime factors
A factor divides exactly into a number. A prime factor is a factor that is also prime. Every whole number greater than 1 that is not prime can be written as a product of prime factors.
Reliable method: keep dividing by prime numbers until the quotient is 1. Use indices to show repeated prime factors. A perfect square has even exponents in its prime factorisation; a perfect cube has exponents divisible by three.
Two reliable methods
Factor tree
- Split the number into any two factors.
- Keep splitting composite factors.
- Stop when every branch ends in a prime.
- Multiply the primes and write repeated factors using indices.
Repeated division
- Start with the smallest prime that divides exactly.
- Keep dividing by prime numbers.
- Continue until the quotient is 1.
- Multiply the prime divisors.
Whichever route you use, the final prime factorisation is the same apart from the order of the factors.
Worked example 4 • Easy
Prime factorisation
Express \(72\) as a product of prime factors.
\[72=8\times9=(2\times2\times2)(3\times3)=2^3\times3^2.\]
Therefore, \(\boxed{72=2^3\times3^2}\).
Using prime exponents for squares and cubes
Prime factorisation is especially useful when a question asks for the smallest multiplier that makes a number a perfect square or perfect cube.
- For a square number, every prime exponent must be even.
- For a cube number, every prime exponent must be a multiple of 3.
Worked example 5 • Medium
Smallest multiplier to make a square
Find the smallest positive integer \(k\) such that \(90k\) is a square number.
Prime factorise 90:
\[90=2\times3^2\times5.\]For a square, each exponent must be even. The powers of 2 and 5 are both 1, so multiply by another 2 and another 5:
\[k=2\times5=10.\]Check:
\[90\times10=900=30^2.\]So \(\boxed{k=10}\).
Worked example 6 • Exam-style
Smallest multiplier to make a cube
Find the smallest positive integer \(m\) such that \(108m\) is a cube number.
Prime factorise:
\[108=2^2\times3^3.\]For a cube, exponents must be multiples of 3. The exponent of 3 is already 3. The exponent of 2 is 2, so one more factor of 2 is needed:
\[m=2.\]Then
\[108\times2=216=6^3.\]Therefore, \(\boxed{m=2}\).
Exam technique
When the question says express as a product of prime factors, finish in prime factors only and use index notation where a prime repeats. Writing \(72=8\times9\) is not a final prime factorisation.
Additional worked example 1 · Medium
Prime factorisation with indices
Write \(540\) as a product of prime factors.
Step 1: \(540=54\times10=(2\times3^3)\times(2\times5)\).
Step 2: Collect repeated factors: \(\boxed{540=2^2\times3^3\times5}\).
Additional worked example 2 · Exam-Style
Make a perfect cube
Find the smallest positive integer \(k\) for which \(72k\) is a perfect cube.
Step 1: \(72=2^3\times3^2\).
Step 2: For a cube, each prime exponent must be a multiple of 3.
Step 3: Multiply by one extra 3, so \(\boxed{k=3}\) and \(72k=216=6^3\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
2.1
Easy [1 mark]
Express \(84\) as a product of prime factors.
Show Answer
Step 1: \(84=2\times42=2\times2\times21\).
Step 2: \(\boxed{2^2\times3\times7}\).
2.2
Medium [2 marks]
Write \(360\) in prime factor form.
Show Answer
Step 1: \(360=36\times10=(2^2\times3^2)(2\times5)\).
Step 2: \(\boxed{2^3\times3^2\times5}\).
2.3
Medium [2 marks]
Find the smallest integer multiplying \(75\) to make a perfect square.
Show Answer
Step 1: \(75=3\times5^2\).
Step 2: 3 has an odd exponent; multiply by another 3.
Step 3: \(\boxed{k=3}\): \(225=15^2\).
2.4
Exam-Style [3–4 marks]
Find the smallest integer \(k\) such that \(108k\) is a perfect cube.
Show Answer
Step 1: \(108=2^2\times3^3\).
Step 2: The exponent of 2 needs one more factor of 2.
Step 3: \(\boxed{k=2}\), making \(216=6^3\).
Choose the operation: use HCF when splitting items into the largest identical groups; use LCM for the earliest time repeating events coincide. In prime-factor form use the smaller exponents for HCF and the larger exponents for LCM.
3
Subtopic 3
Common factors and common multiples
Highest common factor (HCF)
The largest number that is a factor of every number being considered.
Think: largest equal group or largest equal size that divides exactly.
Lowest common multiple (LCM)
The smallest positive number that is a multiple of every number being considered.
Think: the first time repeating cycles or schedules coincide.
Prime-factor method
Write every number as a product of prime factors first.
For HCF
Use only primes common to all the numbers, with the lowest power that appears in all of them.
For LCM
Use every prime needed, with the highest power appearing in any of the numbers.
Reasonableness check: the HCF cannot be larger than the smallest number; the LCM cannot be smaller than the largest number.
Worked example 7 • Easy
HCF and LCM of two numbers
Find the HCF and LCM of 48 and 72.
Prime factorise:
\[48=2^4\times3,\qquad72=2^3\times3^2.\]HCF: take the common primes with the lower powers:
\[\text{HCF}=2^3\times3=24.\]LCM: take all primes with the higher powers:
\[\text{LCM}=2^4\times3^2=144.\]So \(\boxed{\text{HCF}=24}\) and \(\boxed{\text{LCM}=144}\).
Worked example 8 • Medium
HCF and LCM with larger numbers
Find the HCF and LCM of 360 and 504.
\[360=2^3\times3^2\times5\] \[504=2^3\times3^2\times7\]
Common prime factors with the lowest powers:
\[\text{HCF}=2^3\times3^2=72.\]All required prime factors with the highest powers:
\[\text{LCM}=2^3\times3^2\times5\times7=2520.\]Therefore, \(\boxed{72}\) and \(\boxed{2520}\).
Worked example 9 • Exam-style
Largest identical square tiles
A rectangular floor measures 168 cm by 126 cm. It is covered exactly with the largest possible identical square tiles. Find the side length of each tile.
The side length must divide exactly into both 168 and 126, and it must be as large as possible. So this is an HCF problem.
\[168=2^3\times3\times7,\qquad126=2\times3^2\times7.\]Use the common primes with the lowest powers:
\[\text{HCF}=2\times3\times7=42.\]The largest square tile has side length \(\boxed{42\text{ cm}}\).
Worked example 10 • Exam-style
Repeating events
Two warning lights flash together at 09:00. One flashes every 18 seconds and the other every 24 seconds. When will they next flash together?
We need the first time both repeating cycles coincide, so use the LCM.
\[18=2\times3^2,\qquad24=2^3\times3.\] \[\text{LCM}=2^3\times3^2=72\text{ seconds}.\]72 seconds is 1 minute 12 seconds. Therefore, they next flash together at \(\boxed{09{:}01{:}12}\).
Choosing HCF or LCM
Usually HCF
- largest possible equal size
- largest identical groups
- divide exactly with no remainder
Usually LCM
- next time events occur together
- smallest common cycle
- repeating schedules
Additional worked example 1 · Medium
HCF and LCM from primes
Find the HCF and LCM of \(72\) and \(90\).
Step 1: \(72=2^3\times3^2\), \(90=2\times3^2\times5\).
Step 2: HCF \(=2^1\times3^2=\boxed{18}\).
Step 3: LCM \(=2^3\times3^2\times5=\boxed{360}\).
Additional worked example 2 · Exam-Style
Largest identical square tiles
A floor measures \(252\text{ cm}\) by \(180\text{ cm}\). Find the greatest square-tile side length with no cutting, and how many tiles are needed.
Step 1: HCF of \(252\) and \(180\) is \(36\); the largest tile side is \(\boxed{36\text{ cm}}\).
Step 2: Tiles along each side: \(252\div36=7\), \(180\div36=5\).
Step 3: Total \(=7\times5=\boxed{35\text{ tiles}}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
3.1
Easy [1 mark]
Find the HCF of \(24\) and \(36\).
Show Answer
Step 1: Common factors include 1, 2, 3, 4, 6, 12.
Step 2: Highest is \(\boxed{12}\).
3.2
Medium [2 marks]
Find the LCM of \(18\) and \(30\).
Show Answer
Step 1: \(18=2\times3^2\), \(30=2\times3\times5\).
Step 2: Use the greatest powers: \(2\times3^2\times5=\boxed{90}\).
3.3
Medium [2 marks]
Lights flash every 16 s and 24 s. They flash together now. How many seconds until they next coincide?
Show Answer
Step 1: \(16=2^4\); \(24=2^3\times3\).
Step 2: LCM \(=2^4\times3=\boxed{48\text{ s}}\).
3.4
Exam-Style [3–4 marks]
One bus leaves every 28 minutes and another every 42 minutes. Both leave at 08:15. When do they next depart together?
Show Answer
Step 1: LCM of 28 and 42: \(28=2^2\times7\), \(42=2\times3\times7\), so LCM \(=84\) min.
Step 2: 84 min = 1 h 24 min.
Step 3: 08:15 + 1:24 = \(\boxed{09{:}39}\).
✓
Practice
Chapter 1 exercises
Start with the confidence questions, then move to the exam-style problems. Try each question before opening Show Answer. Every answer includes enough working to check your method as well as the final result.
A. Build confidence
1Number types[4 marks]
From the list \(-9,\ 0,\ 14,\ 0.6,\ \sqrt{81},\ \sqrt{11}\):
- write down all the integers;
- write down all the irrational numbers.
Show Answer
a) Integers are whole numbers. Also \(\sqrt{81}=9\).
Integers: \(\boxed{-9,\ 0,\ 14,\ \sqrt{81}}\).
b) \(\sqrt{11}\) is not the root of a perfect square, so it is irrational.
Irrational: \(\boxed{\sqrt{11}}\).
2Rational or irrational?[4 marks]
State whether each number is rational or irrational and give a brief reason.
- \(2.75\)
- \(0.181818\ldots\)
- \(\sqrt{121}\)
- \(5\pi\)
Show Answer
a) Rational — it terminates. \(2.75=\frac{11}{4}\).
b) Rational — it recurs.
c) Rational — \(\sqrt{121}=11\).
d) Irrational — \(\pi\) is irrational and multiplying it by the non-zero rational number 5 does not make it rational.
3Reciprocals[3 marks]
Write the reciprocal of each number.
- \(8\)
- \(\frac{5}{12}\)
- \(-\frac34\)
Show Answer
Flip the number as a fraction:
a) \(\boxed{\frac18}\)
b) \(\boxed{\frac{12}{5}}\)
c) \(\boxed{-\frac43}\)
4Prime factors[4 marks]
Express \(180\) as a product of prime factors in index form.
Show Answer
Break 180 into factors:
\[180=18\times10=(2\times3^2)(2\times5).\]Collect equal primes:
\[\boxed{180=2^2\times3^2\times5}.\]B. Develop your method
5Square multiplier[4 marks]
Find the smallest positive integer \(k\) such that \(150k\) is a square number.
Show Answer
\[150=2\times3\times5^2.\]
For a square, all prime exponents must be even. The powers of 2 and 3 are each 1, so multiply by \(2\times3\).
\[k=6.\]Check: \(150\times6=900=30^2\).
\(\boxed{k=6}\)
6Cube multiplier[4 marks]
Find the smallest positive integer \(m\) such that \(200m\) is a cube number.
Show Answer
\[200=2^3\times5^2.\]
The exponent of 2 is already a multiple of 3. The exponent of 5 is 2, so one more factor of 5 is needed.
\[m=5.\]Check: \(200\times5=1000=10^3\).
\(\boxed{m=5}\)
7HCF and LCM[6 marks]
Find the HCF and LCM of 84 and 126. Show prime-factor working.
Show Answer
\[84=2^2\times3\times7\] \[126=2\times3^2\times7\]
HCF uses common primes with the lower powers:
\[\text{HCF}=2\times3\times7=\boxed{42}.\]LCM uses the highest required powers:
\[\text{LCM}=2^2\times3^2\times7=\boxed{252}.\]8Three numbers[6 marks]
Find the HCF and LCM of 24, 36 and 90.
Show Answer
\[24=2^3\times3,\qquad36=2^2\times3^2,\qquad90=2\times3^2\times5.\]
HCF: common to all three with lowest powers:
\[\text{HCF}=2\times3=\boxed{6}.\]LCM: all primes with highest powers:
\[\text{LCM}=2^3\times3^2\times5=\boxed{360}.\]C. Exam-style challenge
9Irrational number reasoning[3 marks]
Write down an irrational number between 8 and 9. Show clearly why your number lies in this interval.
Show Answer
Since \(8^2=64\) and \(9^2=81\), choose a non-square integer between 64 and 81, such as 70.
\[64<70<81\Rightarrow8<\sqrt{70}<9.\]70 is not a perfect square, so \(\sqrt{70}\) is irrational.
One answer is \(\boxed{\sqrt{70}}\).
10Largest square pieces[5 marks]
A rectangular sheet of card measures 300 mm by 225 mm. It is cut into identical square pieces with no card left over. Find the greatest possible side length of each square.
Show Answer
The side length must divide exactly into both dimensions, and it must be the greatest possible. Find the HCF.
\[300=2^2\times3\times5^2\] \[225=3^2\times5^2\] \[\text{HCF}=3\times5^2=75.\]Greatest possible side length: \(\boxed{75\text{ mm}}\).
11Repeating schedules[5 marks]
A shuttle leaves a station every 28 minutes and another shuttle leaves every 42 minutes. They both leave at 07:30. At what time will they next leave together?
Show Answer
Find the LCM because the departures repeat.
\[28=2^2\times7,\qquad42=2\times3\times7\] \[\text{LCM}=2^2\times3\times7=84\text{ minutes}.\]84 minutes = 1 hour 24 minutes.
\(07{:}30+1{:}24=\boxed{08{:}54}\).
12Combined number reasoning[7 marks]
Two numbers are 360 and 756.
- Express each number as a product of prime factors.
- Find their HCF.
- Find their LCM.
Show Answer
a)
\[360=2^3\times3^2\times5\] \[756=2^2\times3^3\times7\]b) HCF: use common primes with the lower powers:
\[2^2\times3^2=4\times9=\boxed{36}.\]c) LCM: use every prime with the highest power:
\[2^3\times3^3\times5\times7=8\times27\times35=\boxed{7560}.\]Chapter 1 high-grade checklist
Definitions
I can distinguish natural, integer, rational, irrational, prime, square and cube numbers.
Prime factors
I can give a final answer using prime factors and index notation.
Square/cube reasoning
I can use prime exponents to find the smallest multiplier.
HCF vs LCM
I can decide which one a word problem needs and show the prime-factor method.
Chapter 1 revision: This chapter follows the subtopic headings and core concepts in the supplied Cambridge O Level Mathematics, Second Edition. Figure crops come from the provided textbook. Additional questions are original exam-style practice, not reproduced Cambridge examination questions.