Mathematics (Core)Exam code: 0580
Types of Numbers
Number
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