Types of
number.
From number families to prime factors. Learn to recognise the structure behind every number.
Classify numbers · use prime factors · find HCF and LCM · find reciprocals · read large numbers
The number family
Imagine counting books, recording a temperature below zero, and sharing a cake. These situations all use numbers, but they do not all need the same kind of number. A number family groups together numbers with a particular property.
Begin with the counting numbers 1, 2, 3 and so on. Add zero and negative whole numbers to get the integers. Fractions extend the choices further. The families overlap: belonging to one family does not stop a number belonging to another.
A number can belong to more than one family. Start with what it means, then decide where it belongs.
Convention in these notes: natural numbers start at 1. Some texts include 0; follow any definition given in a question.
Rational decimals
They either terminate (end), like 0.625 = 58, or recur (repeat), like 0.272727… = 311.
Irrational decimals
They neither terminate nor recur. A rounded calculator display alone cannot tell you whether a number is irrational.
9 = 3². Its factors are 1, 3 and 9, so it is not prime.
Pause and explainCan 6 belong to more than one number family?
Yes. It is a natural number, an integer and a rational number, because it can be written as 61.
Squares, cubes & primes
Arrange 9 counters in three equal rows of three. The total is 3 × 3, so 9 is a square number. A cube number comes from three equal factors, such as 2 × 2 × 2 = 8.
Prime numbers answer a different question: how many positive whole-number divisors does the number have? Try sharing 7 counters into equal groups. Only groups of 1 or 7 work without a remainder. Keep this divisor test separate from recognising squares and cubes.
Square = n × n
A 4 by 4 array has 16 dots, so 4² = 16.
Cube = n × n × n
A cube with side 3 has volume 3³ = 27.
Primes have exactly two positive factors
A prime number is an integer greater than 1 with only 1 and itself as positive factors. A composite number is an integer greater than 1 with more than two positive factors.
The first ten primes. Every prime after 2 is odd, but not every odd number is prime: 9 = 3 × 3.
Pause and explainWhy is 9 not prime?
It has the positive factors 1, 3 and 9, so it has three factors rather than exactly two.
Break it into prime factors
A factor is a number that divides another number exactly. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with no remainder. Factorising means rewriting a number as a multiplication of factors.
Prime factorisation keeps breaking composite factors into smaller factors until every factor is prime. Start with any convenient factor pair; the final collection of prime factors will be the same. Indices then provide a short way to record repeated factors.
A factor divides a number exactly. A prime factorisation writes a positive integer greater than 1 as a product of primes.
Express 72 as a product of its prime factors.
- Split 72 into any factor pair: 8 × 9.
- Split composite factors again: 8 = 2 × 4, 9 = 3 × 3, and 4 = 2 × 2.
- Collect the prime leaves: 2 × 2 × 2 × 3 × 3.
- Use index notation: 72 = 2³ × 3².
Check: 8 × 9 = 72. A different starting pair, such as 6 × 12, gives the same primes.
Quick divisibility checks
2: last digit is even. 3: digit sum is divisible by 3. 5: last digit is 0 or 5. 9: digit sum is divisible by 9. 10: last digit is 0.
Pause and explainIf 18 = 2 × 9, have we finished its prime factorisation?
No. Nine is not prime. Replace 9 by 3 × 3, giving 18 = 2 × 3².
HCF & LCM: shared, or next together?
Suppose two ribbons are 12 cm and 18 cm long. You want equal-length pieces, with no ribbon left over. A possible piece length must divide both original lengths: it must be a common factor.
The highest common factor gives the largest such piece. A common multiple answers a different kind of question: when two repeating patterns next meet. List a few factors or multiples first so that the prime-factor methods below have a meaning, rather than becoming a rule to memorise.
Highest common factor
The largest positive integer that divides both numbers exactly. Think: largest equal group size.
Factors of 12: 1, 2, 3, 4, 6, 12.
Factors of 18: 1, 2, 3, 6, 9, 18.
Common factors: 1, 2, 3, 6.
HCF = 6.
Lowest common multiple
The smallest positive integer that is a multiple of both numbers. Think: first time cycles meet again.
Multiples of 12: 12, 24, 36, 48, …
Multiples of 18: 18, 36, 54, …
Common multiples: 36, 72, 108, …
LCM = 36.
Find the HCF and LCM of 72 and 120.
72 = 2³ × 3² and 120 = 2³ × 3 × 5
HCF: take only the shared factors.
Use the smaller power of each shared prime.
2³ × 3 = 24
LCM: take everything, counting the shared part once.
Use the greatest power of every prime present.
2³ × 3² × 5 = 360
Check for two positive integers: HCF × LCM = product of the numbers. Here 24 × 360 = 72 × 120 = 8640.
Pause and explainFor 12 and 18, is 36 a common factor or a common multiple?
A common multiple: 36 is 3 × 12 and 2 × 18. It is too large to divide either original length into pieces.
Reciprocals & reading large numbers
The reciprocal of a non-zero number is its multiplicative partner: multiplying the two gives 1. Since 4 × 14 = 1, the reciprocal of 4 is 14. This idea will explain division by fractions later.
Reading a large number is a place-value task. Each step to the left makes a digit worth ten times as much. Group digits into thousands, millions and billions, and read each group before joining the words. A zero can hold a place without being spoken individually.
A reciprocal makes the product 1
For any non-zero number x, its reciprocal is 1x. For a non-zero fraction ab, swap the numerator and denominator.
Read digits in groups of three
Starting from the right, group the digits into units, thousands, millions and billions. Each group has three places: hundreds, tens and units.
6 000 000 000 = six billion. The zero groups hold places; you do not read them aloud.
Write 10 007 in words.
Split it as 10 | 007: ten thousands and seven units.
Going the other way: “four million, twenty thousand and nine” = 4 020 009. Fill missing places with zeros.
Pause and explainWhy does zero have no reciprocal?
There is no number which gives 1 when multiplied by zero; every product with zero is zero.
Your turn. Explain the why.
Before practising, explain the difference between a factor and a multiple without looking at a formula. Then classify one number in several ways: for example, 16 is a natural number, an integer, a rational number and a square.
For each question, decide which test it needs: membership of a family, exact division, repeated multiplication or place value. Write that reason beside your answer. If you get stuck, return to the concrete examples rather than guessing a label.
Try each question before opening its answer. A correct answer is stronger when you can explain the reason.
From −5, 0, 1, 2, 9, √16, √5 and 0.6, identify all the prime numbers and all the irrational numbers.
Reveal worked answer
Express 180 as a product of prime factors. Hence find the HCF and LCM of 72 and 180.
Reveal worked answer
Find the reciprocals of −212 and 0.04.
Reveal worked answer
Write 3 005 012 in words, and write “seven billion and forty” in figures.
Reveal worked answer
Is 64 a square number, a cube number, or both?