Multiples, Factors & Primes (Edexcel IGCSE Maths A (Modular)): Revision Note

Multiples

What are multiples?

• A multiple of a given integer is a number that can be formed by multiplying the integer by another positive integer

  • For example, 12 is a multiple of 3 because 12 = 3 × 4

• Multiples can be considered as the numbers in a times table

• However multiples go beyond times tables and continue forever

  • For example, the multiples of 3 are 3, 6, 9, 12, 15, ..., 300, ..., 3000, ..., 34 567 896, ... 

• Every non-zero number has an infinite number of multiples

• A common multiple of two or more integers is a number that is a multiple of the integers

  • For example, 12 is a common multiple of 4 and 6 6

• Even numbers (2, 4, 6, 8, 10, ...) are multiples of 2

• Odd numbers (1, 3, 5, 7, 9, ...) are not multiples of 2

• Multiples can be algebraic

  • For example, the multiples of  would be

How do I find the multiples of a number?

• Starting with a particular value, multiples can be listed by counting up in steps of that particular value

  • e.g. the multiples of 7 start with 7, then counting up in 7's will give 14, 21, 28, 35 and so on 

• Multiples form a sequence

  • e.g.  7, 14, 21, 28, 35, ...

• Questions may ask you to state the multiples of a value between certain numbers

  • e.g.  the multiples of 7 between 10 and 40 are 14, 21, 28 and 35

Factors

What are factors?

• A factor of a given integer is a value that divides the given number exactly with no remainder

  • 6 is a factor of 18 because 18 ÷ 6 = 3

• Every integer greater than 1 has at least two factors

  • The integer itself, and 1

• A common factor of two or more integers is a number that is a factor of the integers

  • For example, 3 is a common factor of both 21 and 18

How do I find factors?

• Finding all the factors of a particular value can be done by finding factor pairs

• For example when finding the factors of 18

  • 1 and 18 will be the first factor pair

  • Divide by 2, 3, 4 and so on to test if they are factors

    • 18 ÷ 2 = 9, so 9 and 2 are factors

    • 18 ÷ 3 = 6, so 6 and 3 are factors

    • 18 ÷ 4 = 4.5, so 4 is not a factor

    • 18 ÷ 5 = 3.6, so 5 is not a factor

    • 18 ÷ 6 would be next, but we have already found that 6 was a factor

    • So we have now found all the factors of 18: 1, 2, 3, 6, 9, and 18

How do I find factors without a calculator?

• Use a divisibility test

  • Some tests are easier to remember, and more useful, than others

• Once you know that the number has a particular factor, you can divide by that factor to find the factor pair

• If a number is not divisible by an integer, then the number is also not divisible by any multiple of that integer

  • 119 is not divisible by 3

    • so 119 is not divisible by 6, 9, 12, etc, ...

• If a number is divisible by two integers, and the only common factor of those integers is 1, then the number is also divisible by the product of the integers

  • 594 is divisible by 2 and 3

  • The only common factor of 2 and 3 is 1

    • so 594 is divisible by 2×3=6

• Instead of a divisibility test, you could use a formal written method to divide by a value

  • If the result is an integer; you have found a factor

Prime numbers

What are prime numbers?

• A prime number is a number which has exactly two (distinct) factors; itself and 1

  • The first 10 prime numbers are

    • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

    • You should remember at least the first ten prime numbers

• 1 is not a prime number, there are a few reasons for this such as

  • by definition, prime numbers are integers greater than or equal to 2

  • 1 only has one factor

• 2 is the only even prime number

• If a number has any factors other than itself and 1, it is not a prime number

  • For example, 27 is often mistaken for a prime number