Multiples, Factors & Primes (Cambridge (CIE) IGCSE International Maths)

Multiples

What are multiples?

• A multiple is a number which can be divided by another number, without leaving a remainder

  • For example, 12 is a multiple of 3

    • 12 divided by 3 is exactly 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 is multiple that is shared by more than one number

  • For example, 12 is a common multiple of 4 and 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 number is a value that divides the given number exactly, with no remainder

  • 6 is a factor of 18

    • because 18 divided by 6 is exactly 3

• Every integer greater than 1 has at least two factors

  • The integer itself, and 1

• A common factor is a factor that is shared by more than one number

  • 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

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

• 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

How do I test for divisibility by 2?

• A number is divisible by 2 if the last digit is even (a multiple of 2)

  • 126
    6 is even so 126 is divisible by 2

  • 135
    5 is odd so 135 is not divisible by 5

How do I test for divisibility by 3?

• A number is divisible by 3 if the sum of the digits is divisible by 3 (a multiple of 3)

  •  123
    1 + 2 + 3 = 6;  6 is a multiple of 3, so 123 is divisible by 3

  • 134
    1 + 3 + 4 = 8; 8 is not a multiple of 3, so 134 is not divisible by 3

How do I test for divisibility by 4 or 8?

• A number is divisible by 4 if halving the number twice results in an integer

  • 128
    128 ÷ 2 = 64;  64 ÷ 2 = 32;  32 is an integer so 128 is divisible by 4

  • 134
    134 ÷ 2 = 67;  67 ÷ 2 = 33.5;  33.5 is not an integer so 134 is not divisible by 4

• A number is divisible by 8 if it can be halved 3 times and the result is an integer

How do I test for divisibility by 5 or 10?

• A number is divisible by 5 if the last digit is a 0 or 5

  • 165
    The last digit is 5;  165 is divisible by 5

  • 230
    The last digit is 0;  230 is divisible by 5

  • 162
    The last digit is 2;  162 is not divisible by 5

• A number is divisible by 10 if the last digit is a 0

What are some other divisibility tests?

• The following are harder divisibility tests

• You don't need to remember them, but they can speed up your working

• You could instead use a formal written method to carry out a division instead

• A useful fact is that if a value is divisible by two numbers, it is also divisible by the product of those two numbers

  • e.g. A number is divisible by 6 if it is divisible by both 2 and 3

  • A number is divisible by 12 if it is divisible by both 4 and 3

• A number is divisible by 7 if you get a multiple of 7 when you double the last digit, and subtract it from the remaining part of the number

  • 245

    • Double 5 is 10

    • 24 - 10 = 14 which is a multiple of 7, so 245 is a multiple of 7

  • 906

    • Double 6 is 12

    • 90-12=78 which is not a multiple of 7, so 906 is not a multiple of 7

• A number is divisible by 9 if the sum of the digits is divisible by 9 (similar to the rule for 3)

• A number is divisible by 11 if you get an answer of 0 or a multiple of 11 when you alternately add and subtract the digits

  • 1364

    • +1-3+6-4=0   so 1364 is a multiple of 11

  • 428

    • +4-2+8=10   so 428 is not a multiple of 11

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