1. Types of number
Integers and natural numbers
- The integers run in both directions from zero without fractional parts: … −3, −2, −1, 0, 1, 2, 3 …
- Counting from 1 upwards gives the natural numbers 1, 2, 3, 4, …, a set that leaves zero out
- A number written as a decimal or a fraction is only an integer if it simplifies to a whole number
- 18/3 is an integer because it equals 6
Multiples
- A multiple of an integer is the result of multiplying it by a positive integer
- The multiples of 7 are 7, 14, 21, 28, 35, …
- A common multiple of two integers is a multiple of both
- 24 is a common multiple of 6 and 8
- Multiples can be written algebraically: the multiples of n are n, 2n, 3n, 4n, …
Factors
- A factor of an integer divides into it exactly, leaving no remainder
- Beyond 1, no integer has fewer than two factors, since 1 and the number itself always divide it
- Factors are found in pairs, which is why testing only up to the square root is enough
- For 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6, giving factors 1, 2, 3, 4, 6, 8, 12, 24
- A common factor of two integers is a factor of both
- 5 is a common factor of 20 and 35
Divisibility tests
- Divisibility tests decide whether a number is a factor without dividing, which matters on the non-calculator papers
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is even | 4 918 |
| 3 | Digit sum is a multiple of 3 | 5 121, since 5+1+2+1 = 9 |
| 4 | Last two digits form a multiple of 4 | 7 316, since 16 ÷ 4 = 4 |
| 5 | Last digit is 0 or 5 | 8 470 |
| 6 | Passes both the 2 test and the 3 test | 2 934 |
| 8 | Last three digits form a multiple of 8 | 9 512, since 512 ÷ 8 = 64 |
| 9 | Digit sum is a multiple of 9 | 6 741, since 6+7+4+1 = 18 |
| 10 | Last digit is 0 | 3 250 |
Primes, squares, cubes and roots
- A prime number has exactly two factors, 1 and itself
- Among the primes only 2 is even, since every other even number has 2 as a third factor
- 1 is not prime, because it has only one factor
- A square number is an integer multiplied by itself: 1, 4, 9, 16, 25, 36, …
- A cube number is an integer multiplied by itself twice: 1, 8, 27, 64, 125, …
- The square root of a number is the value that squares to give it, written √
- √49 = 7, because 7² = 49
- The cube root reverses cubing, written ∛
- ∛125 = 5, because 5³ = 125
Exam tip
No extras in a list of factors
When a question asks for all the factors of a number, or all the common factors of two numbers, the complete list earns both marks. One factor missing, or one number too many, drops it to one mark. Find the factors in pairs so that none is missed, then check each one divides exactly: 1, 2, 4, 5, 8, 10, 15, 20, 40 for the factors of 40 scores one mark, because 15 is not a factor.
Worked example
Factors, multiples and primes
Write down all the factors of 36. State which of them are prime, and write down the first three common multiples of 4 and 6.
Solution:
- Find the factors in pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
- The pairs stop once they meet, at 6 × 6, so nothing beyond 6 needs testing
- The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36
- Test each for exactly two factors: 1 has one factor, 4 has three, 9 has three
- The prime factors are 2 and 3
- For the common multiples, list each set: 4, 8, 12, 16, 20, 24, 28, 32, 36 and 6, 12, 18, 24, 30, 36
- The first three values in both lists are 12, 24 and 36
2. Rational and irrational numbers
Rational numbers
- Any value expressible as one integer over another, with a non-zero denominator, is a rational number
- Every integer is rational, because it can be written over a denominator of 1
- Every terminating decimal is rational
- 0.35 = 7/20
- Every recurring decimal is rational
- 0.333… = 1/3
Irrational numbers
- An irrational number admits no such integer-over-integer form
- Its decimal form never terminates and never settles into a repeating block
- The square root of a positive integer that is not a square number is irrational
- √7 is irrational, but √36 is not, because it equals 6
- π is irrational
- A non-zero rational number multiplied by an irrational number is still irrational
- 5√2 and π/4 are both irrational
Worked example
Naming an irrational number in a range
Write down an irrational number between 3 and 4. Explain how you know it is irrational.
Solution:
- Square both ends of the range: 3² = 9 and 4² = 16
- Any square root of a number between 9 and 16 therefore lies between 3 and 4
- Choose a value in that range that is not a square number, such as 10
- √10 is one possible answer, and √11, √12, √13, √14 and √15 all work equally well
- It is irrational because 10 is not a square number, so its square root neither terminates nor recurs
- √16 would not do, because 16 is a square number and √16 = 4, which is rational
Worked example
Picking numbers from a list
Here is a list of numbers: 21, 29, 36, 57, 64, 75, √30, √144. From the list, write down a prime number, a square number, a cube number, a factor of 150 and an irrational number.
Solution:
- Test each number in the list for the property asked, because some numbers that look prime are not
- 21 = 3 × 7, and the digits of 57 add to 12, a multiple of 3, so 57 = 3 × 19
- 36 and 64 are even, 75 ends in 5, and √144 = 12, so none of them is prime
- 29 is not divisible by 2, 3 or 5, and 6 × 6 = 36 is already more than 29, so no larger factor needs testing: the prime number is 29
- 36 = 6 × 6 and 64 = 8 × 8, so the square number is 36 or 64: either one answers the question
- 64 = 4 × 4 × 4, so the cube number is 64; 36 is not a cube, since 3³ = 27 and 4³ = 64
- 150 = 2 × 75, so the factor of 150 is 75
- 30 is not a square number, since 5² = 25 and 6² = 36, so the irrational number is √30
- √144 is rational, because it equals 12
3. Reciprocals
Finding a reciprocal
- Inverting a number gives its reciprocal: 1 is divided by the number, and the result is what the original must be multiplied by to reach 1
- Every non-zero value and its reciprocal have a product of exactly 1
- For a fraction, numerator and denominator swap places
- Inverting 3/8 gives 8/3
- The reciprocal of an integer n is 1/n
- The reciprocal of 5 is 1/5
- A decimal is easiest to invert by first writing it as a fraction
- 0.4 = 2/5, so its reciprocal is 5/2 = 2.5
- Zero has no reciprocal, because division by zero is undefined
- In index form the reciprocal of a is a⁻¹
Exam tip
Give the reciprocal as a value
Reciprocals were set in 9 of the 22 papers. When a question asks for the value of the reciprocal of a decimal, writing 1/0.625 or 1 ÷ 0.625 earns only one of the two marks: the second is for the value. Write the decimal as a fraction, turn it over and simplify, so 0.625 = 5/8 gives 8/5 = 1.6.
Worked example
Finding the reciprocal of a mixed number
Write down the reciprocal of 2 1/4.
Solution:
- Convert the mixed number to a top-heavy fraction: 2 1/4 = 9/4
- Swap numerator and denominator: the reciprocal is 4/9
- Check: 9/4 × 4/9 = 36/36 = 1