Maths · 1 Number

1.1 Types of number

Before you start

Times tables up to $12 \times 12$ and short division, without a calculator.
Place value: units, tens, hundreds, thousands, and how a zero holds an empty place.
Negative numbers on a number line, e.g. $- 3$ is less than $1$ .
What a fraction means, and turning a mixed number such as $2 \frac{1}{2}$ into the improper fraction $\frac{5}{2}$ .

Syllabus objectives covered (1)

Identify and use: • natural numbers (0, 1, 2, …) • integers (positive, zero and negative) • prime numbers • square numbers • cube numbers • triangle numbers • common factors • common multiples • rational and irrational numbers • reciprocals.

Part 1 of 6

Natural numbers, integers, and numbers in words

The natural numbers are the counting numbers, and in this syllabus they start at zero: $0, 1, 2, 3, \dots$ . Some books start at 1. Use the syllabus definition, because that is the one your paper is written against.
The integers are the whole numbers in both directions: $\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots$ . Every natural number is an integer. A negative whole number such as $- 5$ is an integer but not a natural number.
Decide by the value, not by how the number is written. $\sqrt{49}$ is $7$ and $\frac{12}{4}$ is $3$ , so both are natural numbers. $\frac{7}{2}$ is $3.5$ , so it is not an integer at all.
To read or write a large number, split the digits into groups of three from the right. Cambridge separates the groups with a space, not a comma: $6 000 000$ , not 6,000,000.
Know the size of each group name. A thousand has 3 zeros, a million has 6 zeros, and a billion has 9 zeros.
$1 billion = 1 000 000 000 = 1000 million$
Read each group of three as an ordinary number, then say its name. $3 040 205$ is 'three million, forty thousand, two hundred and five'.
The trap is the empty place. 'Fifty thousand and nine' has nothing in its hundreds or tens, so you must write zeros there: $50 009$ , not $509$ or $5009$ . Write the full set of groups first, then fill them in.

Part 2 of 6

Prime numbers and prime factors

A prime number has exactly two factors: $1$ and itself. $7$ is prime because only $1 \times 7$ makes it.
Exactly two factors means $1$ is not prime, because it has only one factor. And $2$ is prime, the only even prime, because every other even number also divides by 2.
The primes below 50 are worth knowing on sight: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$ .
To test a larger number, divide it by the primes $2, 3, 5, 7, \dots$ in turn. You only need to go up to its square root. $91 = 7 \times 13$ , so 91 is not prime. For 97, $\sqrt{97}$ is just under 10, and none of 2, 3, 5 or 7 divides it, so 97 is prime.
Every whole number greater than 1 that is not prime can be written as a product of primes. There is only one way to do it, apart from the order. This product is called its prime factorisation.
To find it, divide by the smallest prime that works, and repeat until you reach 1. For 72: $72 \div 2 = 36$ , $36 \div 2 = 18$ , $18 \div 2 = 9$ , $9 \div 3 = 3$ , $3 \div 3 = 1$ . The primes you divided by are the factorisation.
$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^{3} \times 3^{2}$
Write the final answer as a product. Index form, $2^{3} \times 3^{2}$ , is the neatest way, and $2 \times 2 \times 2 \times 3 \times 3$ is also a correct product of primes. A factor tree is good working, but it is not the answer. $8 \times 9$ is not an answer either, because 8 and 9 are not prime.
Check by multiplying back: $8 \times 9 = 72$ . It takes ten seconds and catches a missed factor.

Part 3 of 6

Square, cube and triangle numbers

A square number is a whole number multiplied by itself. The name comes from dots: $4$ rows of $4$ dots make a square of $16$ .
$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$
A cube number is a whole number used three times in a multiplication: $4 \times 4 \times 4 = 64$ .
$1, 8, 27, 64, 125, \dots, 1000 = 10^{3}$
A triangle number is the number of dots in a triangle with rows of 1, 2, 3, … dots. So the triangle numbers are $1$ , $1 + 2 = 3$ , $1 + 2 + 3 = 6$ , and so on. Each one is the previous one plus the next whole number.
$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$
To test a larger number, a shortcut helps: the $n$ th triangle number is $\frac{n (n + 1)}{2}$ . For example, the 10th is $\frac{10 \times 11}{2} = 55$ . You do not have to learn this, but it is faster than adding up a long list.
One number can belong to several types. $1$ is square, cube and triangle. $64$ is square ( $8^{2}$ ) and cube ( $4^{3}$ ). $36$ is square ( $6^{2}$ ) and triangle. When a question asks for a number that is 'both', look for these overlaps.

Part 4 of 6

Common factors, common multiples, HCF and LCM

A factor of a number divides it exactly. A multiple of a number is that number times a whole number. Factors are small and finite: 12 has factors $1, 2, 3, 4, 6, 12$ . Multiples go on forever: $12, 24, 36, \dots$ .
A common factor of two numbers is a factor of both. The common factors of 12 and 18 are $1, 2, 3, 6$ . The largest one, $6$ , is the highest common factor (HCF).
A common multiple is a multiple of both. Multiples of 12 are $12, 24, 36, \dots$ and multiples of 18 are $18, 36, \dots$ . The first one they share, $36$ , is the lowest common multiple (LCM).
Listing works for small numbers. For bigger ones, use prime factors. Write both numbers in index form first.
$60 = 2^{2} \times 3 \times 5 72 = 2^{3} \times 3^{2}$
HCF: take only the primes that appear in both numbers, each at its lower power. Both contain 2 and 3. The lower powers are $2^{2}$ and $3^{1}$ , so the HCF is $2^{2} \times 3 = 12$ . The HCF must divide both numbers, so it cannot use more of a prime than either one has.
LCM: take every prime that appears in either number, each at its higher power. The primes are 2, 3 and 5, so the LCM is $2^{3} \times 3^{2} \times 5 = 360$ . The LCM must contain each number completely, so it needs the most of every prime.
A quick check for two numbers: HCF $\times$ LCM equals the two numbers multiplied together. Here $12 \times 360 = 4320$ and $60 \times 72 = 4320$ .
In a word problem, the wording tells you which one. 'Next happen together' or 'smallest amount that both divide into' is an LCM. 'Largest equal groups' or 'biggest size that fits both exactly' is an HCF.

Part 5 of 6

Rational and irrational numbers

A rational number is one you can write as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
That includes more than it seems. Every integer is rational, because $5 = \frac{5}{1}$ . Every terminating decimal is rational, because $0.35 = \frac{35}{100} = \frac{7}{20}$ . Every recurring decimal is rational too: $0. \dot{3} = 0.333 \dots = \frac{1}{3}$ .
An irrational number cannot be written as a fraction of two integers. Its decimal goes on forever with no repeating pattern. The ones you will meet are $π$ , the square root of any whole number that is not a square number (such as $\sqrt{2}$ or $\sqrt{50}$ ), and the cube root of any whole number that is not a cube number (such as $\sqrt[3]{10}$ ).
A root sign does not make a number irrational. $\sqrt{49} = 7$ and $\sqrt{\frac{9}{4}} = \frac{3}{2}$ are both rational. Always work out the value before you decide.
Your calculator shows $π$ as $3.141592654$ , but that is only a rounded value. $π$ itself never ends. In the same way, $\frac{22}{7}$ is a rational number that is close to $π$ . It is not equal to $π$ .
To give an irrational number between two integers, square the two ends. Between 7 and 8 means between $49$ and $64$ . So $\sqrt{50}, \sqrt{51}, \dots, \sqrt{63}$ all work. Give the root itself, not a rounded decimal. A decimal such as $7.4$ is rational.
Two irrational numbers can combine to give a rational one: $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ . A rational number plus an irrational number is always irrational, e.g. $2 + \sqrt{3}$ . So when you are asked about a product of roots, work it out first and then decide.

Part 6 of 6

Reciprocals

The reciprocal of a number is $1$ divided by that number. The reciprocal of $5$ is $\frac{1}{5}$ .
A number multiplied by its reciprocal always gives 1. That is the definition you can use to check any answer: $5 \times \frac{1}{5} = 1$ .
For a fraction, turn it upside down. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$ , because $\frac{3}{4} \times \frac{4}{3} = 1$ .
For a mixed number, change it to an improper fraction first. $2 \frac{1}{2} = \frac{5}{2}$ , so its reciprocal is $\frac{2}{5}$ . If you turn over only the fraction part, you get a wrong answer.
For a decimal, change it to a fraction or work out $1 \div$ the decimal. $0.25 = \frac{1}{4}$ , so its reciprocal is $4$ .
The sign does not change. The reciprocal of $- \frac{2}{3}$ is $- \frac{3}{2}$ . A negative times a negative is positive, so $- \frac{2}{3} \times (- \frac{3}{2}) = 1$ . A positive reciprocal would give $- 1$ , which is why the sign must stay negative.
Zero has no reciprocal. $\frac{1}{0}$ has no value, because no number multiplied by 0 gives 1.

Worked examples

Example 1. Here is a list of numbers: $- 3, 1, 10, 27, 36, \sqrt{49}, \sqrt{50}, 0. \dot{4}, 2 π$ . From the list, write down (a) the natural numbers, (b) the integers, (c) the prime numbers, (d) the square numbers, (e) the cube numbers, (f) the triangle numbers, (g) the irrational numbers.

Show the working

First, work out the value of anything in disguise. $\sqrt{49} = 7$ . $\sqrt{50}$ is between 7 and 8 and 50 is not a square number, so it is irrational. $0. \dot{4} = 0.444 \dots = \frac{4}{9}$ . $2 π$ is irrational because $π$ is.
(a) Natural numbers start at 0 and have no negatives or fractions: $1, 10, 27, 36, \sqrt{49}$ .
(b) Integers add the negative whole numbers: $- 3, 1, 10, 27, 36, \sqrt{49}$ .
(c) Primes have exactly two factors. $\sqrt{49} = 7$ is prime. $1$ is not prime, because it has only one factor.
(d) Square numbers: $1 = 1^{2}$ and $36 = 6^{2}$ .
(e) Cube numbers: $1 = 1^{3}$ and $27 = 3^{3}$ .
(f) Triangle numbers are $1, 3, 6, 10, 15, 21, 28, 36, \dots$ , so $1, 10$ and $36$ are in the list.
(g) Irrational: $\sqrt{50}$ and $2 π$ . Everything else can be written as a fraction of integers, including $0. \dot{4} = \frac{4}{9}$ .

Answer: (a) $1, 10, 27, 36, \sqrt{49}$ (b) $- 3, 1, 10, 27, 36, \sqrt{49}$ (c) $\sqrt{49}$ (d) $1, 36$ (e) $1, 27$ (f) $1, 10, 36$ (g) $\sqrt{50}, 2 π$

Example 2. Find the highest common factor and the lowest common multiple of $126$ and $180$ .

Show the working

Write each number as a product of primes by repeated division. $126 \div 2 = 63$ , $63 \div 3 = 21$ , $21 \div 3 = 7$ , $7 \div 7 = 1$ .
$126 = 2 \times 3^{2} \times 7$
$180 \div 2 = 90$ , $90 \div 2 = 45$ , $45 \div 3 = 15$ , $15 \div 3 = 5$ , $5 \div 5 = 1$ .
$180 = 2^{2} \times 3^{2} \times 5$
HCF: the primes in both are 2 and 3. Take the lower power of each: $2^{1}$ and $3^{2}$ .
$HCF = 2 \times 3^{2} = 18$
LCM: every prime in either number is 2, 3, 5 or 7. Take the higher power of each: $2^{2}$ , $3^{2}$ , $5$ , $7$ .
$LCM = 2^{2} \times 3^{2} \times 5 \times 7 = 1260$
Check: $18 \times 1260 = 22 680$ and $126 \times 180 = 22 680$ . They match.

Answer: HCF $= 18$ , LCM $= 1260$ .

Example 3. A teacher has 48 pencils and 72 rulers. She puts them into identical packs, using every pencil and every ruler, with no items left over. Work out the largest number of packs she can make, and what is in each pack.

Show the working

Decide which tool the question needs. The number of packs must divide 48 and 72 exactly, and we want the largest such number. That is the HCF.
Write both in prime factors.
$48 = 2^{4} \times 3 72 = 2^{3} \times 3^{2}$
Take the shared primes at the lower power.
$HCF = 2^{3} \times 3 = 24$
Share out: $48 \div 24 = 2$ pencils and $72 \div 24 = 3$ rulers in each pack.
Answer in the context of the question, not just with a number.

Answer: $24$ packs, each with $2$ pencils and $3$ rulers.

Example 4. (a) Write down an irrational number between 4 and 5. (b) Show that $\sqrt{2} \times \sqrt{18}$ is rational. (c) Find the reciprocal of $1 \frac{1}{4}$ .

Show the working

(a) Square the ends: $4^{2} = 16$ and $5^{2} = 25$ . Any square root of a whole number from 17 to 24 lies between 4 and 5, and none of them is a square number, so each one is irrational.
(b) Multiply under one root, then work out the value.
$\sqrt{2} \times \sqrt{18} = \sqrt{36} = 6$
Then say why that answers the question: $6 = \frac{6}{1}$ , a fraction of two integers, so it is rational. The last sentence is the part that shows the result.
(c) Change the mixed number to an improper fraction first: $1 \frac{1}{4} = \frac{5}{4}$ . Turn it over to get $\frac{4}{5}$ .
Check: $\frac{5}{4} \times \frac{4}{5} = 1$ . Correct.

Answer: (a) e.g. $\sqrt{20}$ (any $\sqrt{17}$ to $\sqrt{24}$ ) (b) $\sqrt{36} = 6 = \frac{6}{1}$ , so it is rational (c) $\frac{4}{5}$ $(= 0.8)$

Where people go wrong

Mistake: Counting $1$ as a prime number.

Instead: A prime has exactly two factors. $1$ has only one, so it is not prime. The smallest prime is $2$ .

'Divides only by 1 and itself' seems to include 1, because 1 and itself are the same number. The 'exactly two factors' definition removes that problem.

Mistake: Multiplying the two numbers together to get the LCM, e.g. giving the LCM of 12 and 18 as $216$ .

Instead: $12 \times 18$ is a common multiple, but not the lowest. The LCM is $36$ . Use prime factors and take each prime at its higher power only.

Multiplying counts the shared factors twice. The product equals the LCM only when the two numbers have no common factor except 1.

Mistake: Giving a prime factorisation that still contains a number that is not prime, e.g. $72 = 8 \times 9$ , or writing a list such as '2, 2, 2, 3, 3'.

Instead: Keep dividing until every factor is prime, and write a product: $72 = 2^{3} \times 3^{2}$ .

8 and 9 are factors of 72 but not prime factors. A list is not a product, so it does not answer the question.

Mistake: Thinking every number with a root sign, or every long decimal, is irrational.

Instead: Work out the value first. $\sqrt{49} = 7$ is rational. $0. \dot{3} = \frac{1}{3}$ is rational, because its digits repeat.

Irrational means it cannot be written as a fraction of integers. Many roots and recurring decimals can.

Mistake: Taking the reciprocal of a mixed number by turning over only the fraction part, e.g. $2 \frac{1}{2} \to 2 \frac{2}{1}$ .

Instead: Convert first: $2 \frac{1}{2} = \frac{5}{2}$ , so the reciprocal is $\frac{2}{5}$ .

The reciprocal must multiply the whole number to give 1. Check: $\frac{5}{2} \times \frac{2}{5} = 1$ , but $2 \frac{1}{2} \times 4 = 10$ .

Check yourself

Write fourteen million, two hundred and five thousand in figures.
Answer
$14 205 000$
Write $40 060$ in words.
Answer
Forty thousand and sixty.
Write $150$ as a product of its prime factors.
Answer
$150 = 2 \times 3 \times 5^{2}$ (check: $2 \times 3 \times 25 = 150$ ).
Find the HCF and the LCM of $18$ and $30$ .
Answer
$18 = 2 \times 3^{2}$ and $30 = 2 \times 3 \times 5$ . HCF $= 2 \times 3 = 6$ . LCM $= 2 \times 3^{2} \times 5 = 90$ .
Which of $15$ , $28$ , $45$ and $50$ are triangle numbers?
Answer
$15$ , $28$ and $45$ . The triangle numbers go $\dots, 36, 45, 55$ , so 50 is not one.
Which of these are irrational: $\sqrt{16}$ , $\sqrt{18}$ , $3.14$ , $π$ , $0. \dot{7}$ ?
Answer
$\sqrt{18}$ and $π$ . $\sqrt{16} = 4$ , $3.14 = \frac{314}{100}$ and $0. \dot{7} = \frac{7}{9}$ are all rational.
Write down the reciprocal of $0.2$ and the reciprocal of $- \frac{3}{4}$ .
Answer
$0.2 = \frac{1}{5}$ , so its reciprocal is $5$ . The reciprocal of $- \frac{3}{4}$ is $- \frac{4}{3}$ .
Find every number from 1 to 100 that is both a square number and a cube number.
Answer
$1$ and $64$ . The cubes up to 100 are $1, 8, 27, 64$ , and only $1$ and $64 = 8^{2}$ are also squares.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Here is a list of numbers: $2, 9, 15, 21, 27, 36, 49, 51, 64$ .

(a) Write down the prime number in the list. 1 mark No calculator

Mark scheme

B1 2 cao

(b) Write down the number in the list that is both a square number and a cube number. 1 mark No calculator

Mark scheme

B1 64 cao

(c) Write down the number in the list that is both a square number and a triangle number. 1 mark No calculator

Mark scheme

B1 36 cao

(d) Explain why $51$ is not a prime number. 1 mark No calculator

Mark scheme

B1 51 = 3 × 17 oe, or states that 3 (or 17) is a factor, so 51 has more than two factors

Question 2. You are given that $196 = 2^{2} \times 7^{2}$ .

(a) Write $280$ as a product of its prime factors. 2 marks No calculator

Mark scheme

M1 a correct factor tree or repeated division reaching at least two correct prime factors, or the list 2, 2, 2, 5, 7 seen
A1 2³ × 5 × 7 oe, e.g. 2 × 2 × 2 × 5 × 7, written as a product

(b) Find the highest common factor (HCF) of $280$ and $196$ . 2 marks No calculator

Mark scheme

M1 2² × 7 seen, or a list of factors of both numbers with 28 in each; FT their (a)
A1 28 cao

(c) Find the lowest common multiple (LCM) of $280$ and $196$ . 2 marks No calculator

Mark scheme

M1 2³ × 5 × 7² seen, or 280 × 196 ÷ their HCF; FT their (a)
A1 1960 cao

Question 3. A red light flashes every $24$ seconds. A green light flashes every $40$ seconds. The two lights are switched on together and both flash at that moment.

(a) Work out the number of seconds until the lights next flash at the same time. 2 marks No calculator

Mark scheme

M1 24 = 2³ × 3 and 40 = 2³ × 5, or lists of multiples of 24 and of 40 reaching 120
A1 120 cao

(b) The lights stay on for $25$ minutes. Find the number of times they flash at the same time in those $25$ minutes, including the moment they are switched on. 2 marks No calculator

Mark scheme

M1 25 × 60 ÷ their 120 oe, i.e. 1500 ÷ 120 = 12.5 seen; FT their (a)
A1 13 cao

Question 4.

(a) Write down an irrational number between $7$ and $8$ . 1 mark No calculator

Mark scheme

B1 any √n with n an integer and 50 ≤ n ≤ 63, or another correct irrational number such as π + 4; a rounded decimal is not accepted

(b) Write down the reciprocal of $1 \frac{3}{5}$ . 1 mark No calculator

Mark scheme

B1 5/8 oe, e.g. 0.625

(c) Show that $\sqrt{3} \times \sqrt{12}$ is a rational number. 2 marks No calculator

Mark scheme

M1 √36 seen, or √3 × √12 = √(3 × 12)
A1 = 6, with a statement that 6 = 6/1 (an integer / a fraction of integers) so it is rational; nfww

(d) Explain why $0. \dot{6}$ is a rational number. 1 mark No calculator

Mark scheme

B1 0.6̇ = 2/3 oe, a fraction with integer numerator and denominator; or states that it is a recurring decimal

How this is examined

Using prime factors to find the HCF and LCM of two larger numbers: Two linked parts straight after a prime-factor part, often with the second number's factorisation given in the stem, about 2 marks each HCF: the primes the two numbers share, each at its lower power. LCM: every prime that appears, each at its higher power. The correct product (e.g. $2^{2} \times 7$ ) shows the method; the final value worked out as a single number (e.g. $28$ ) is the answer.
Spotting that a word problem is really an LCM or an HCF: A short context question — lights, bells or buses repeating, or items shared into identical groups — worth 2 or 3 marks 'Next together / at the same time again' means LCM; 'largest number of identical groups' means HCF. Show the prime factors or the lists of multiples, then answer in the context's units ('120 seconds', '24 packs').
Writing a number as a product of its prime factors: A 2-mark 'Write … as a product of its prime factors', usually on the non-calculator paper A final line that is a product of primes, best in index form: $2^{3} \times 5 \times 7$ . A factor tree with no product written, a list such as '2, 2, 2, 5, 7', or a product containing a non-prime such as $8 \times 35$ is not the answer.
Showing that a product of roots is rational, or giving an irrational number in a range: A 1-mark Write down an irrational number between two integers, or a 2-mark Show that a product such as $\sqrt{3} \times \sqrt{12}$ is rational For the range: an exact root such as $\sqrt{55}$ , not a rounded decimal like $7.4$ . For Show that: the working to a single integer or fraction ( $\sqrt{36} = 6$ ), then a final statement that it is a fraction of integers, so it is rational.
Picking numbers of a given type out of a list, including reciprocals: A list of eight or so numbers followed by 1-mark Write down parts: the prime, a cube, the number that is two types at once; or a 1-mark reciprocal of a fraction or decimal Exactly the number asked for and nothing extra — an extra number gives the examiner a wrong answer to mark too. Check that 1 is not prime, that numbers such as 51 and 91 are not prime, and that 36 is both square and triangle.

Words to know

Natural numbers: The counting numbers starting at zero: $0, 1, 2, 3, \dots$
Integers: All whole numbers, positive, zero and negative: $\dots, - 2, - 1, 0, 1, 2, \dots$
Prime number: A whole number with exactly two factors, 1 and itself. 1 is not prime; 2 is the only even prime.
Prime factorisation: A number written as a product of primes, e.g. $72 = 2^{3} \times 3^{2}$ .
Square number: A whole number multiplied by itself: $1, 4, 9, 16, \dots$
Cube number: A whole number used three times in a multiplication: $1, 8, 27, 64, \dots$
Triangle number: The number of dots in a triangle with rows of 1, 2, 3, … dots: $1, 3, 6, 10, 15, \dots$
Factor: A whole number that divides another exactly, e.g. 4 is a factor of 12.
Multiple: A number times a whole number, e.g. 36 is a multiple of 12.
Highest common factor (HCF): The largest number that is a factor of two or more numbers.
Lowest common multiple (LCM): The smallest number, other than zero, that is a multiple of two or more numbers.
Rational number: A number that can be written as $\frac{a}{b}$ with $a$ and $b$ integers and $b \neq 0$ . Includes integers, terminating decimals and recurring decimals.
Irrational number: A number that cannot be written as a fraction of two integers, e.g. $π$ or $\sqrt{2}$ . Its decimal never ends and never repeats.
Reciprocal: 1 divided by a number. A number times its reciprocal is 1. Zero has no reciprocal.

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