Definition
Composite numbers are positive, whole numbers, greater than one, that are not prime. Composite numbers can be expressed as the product of two or more other positive whole numbers. Taken further, composite numbers can be expressed as the product of two or more prime numbers.
Explanation
Understanding composite numbers, involves recognising and understanding prime numbers.
Whilst prime numbers have exactly two distinct factors (one and itself), composite numbers have more than two distinct factors - i.e. at least one factor is neither 1 nor itself.
The smallest composite number is 4. It has factors 1, 2 and 4.
Prime numbers can be seen as building blocks for all other (positive, whole) numbers.
Composite numbers can be written as a product of some of their factors. This can be extended such that composite numbers can be written as a product of prime numbers, and hence we can consider prime numbers as the blocks that build all non-prime (composite) numbers.
For example, the composite number 24 can be written as 2 × 3 × 4 (which are all factors of 24), but 4, which isn’t prime, can be written as 2 × 2.
Hence, the composite number 24 can be written as the product of the primes:
2 × 2 × 2 × 3 (or 23 × 3 using indices)
This building blocks idea can also explain why 1 is not a prime number - multiplying by 1 doesn’t do anything!
Example
Question: Giving a reason for each, state whether the odd whole numbers between 71 and 79 are composite.
Solution:
Step 1: Be clear about which numbers need to be considered
whole, odd, so 71, 73, 75, 77, 79
Step 2: Use recognition and quick divisibility tests to find the composite numbers
75 - ends in the digit 5, so is divisible by 5
77 - follows the pattern for the 11 times table, so is divisible by 11
Step 3: This is actually a shortcut - knowing the prime numbers up to 100
71, 73 and 79 are all prime
(you can also consider that these are not in any times tables)
Step 4: Ensure you are answering the question!
71 is not a composite number as it is a prime number
73 is not a composite number as it is a prime number
75 is divisible by 5 so is a composite number
77 is divisible by 7 so is a composite number
79 is not a composite number as it is a prime number
Common mistakes (and how to avoid them)
Mistake 1: The number 1 is positive and whole, but is neither a prime number nor a composite number
The number 1 causes much confusion when it comes to talking about prime numbers and composite numbers. But the number 1 is neither. The number 1 is a unitary value.
Prime numbers are defined as being integers greater than or equal to 2.
Composite numbers are made from prime numbers, so 1 cannot be a composite number.
How to avoid this: In day-to-day language, mathematicians and teachers will loosely refer to prime numbers, composite numbers, etc without specifying every time they have to be positive integers greater than or equal to 2. To not be confused by the unit number 1, remember that, by definition, it shouldn’t even be considered!
Mistake 2: Composite numbers can be written as products in several different ways
All composite numbers can be written as a product of prime numbers. You will see this process called prime factorisation, or expressing a number as a product of its prime factors. However, composite numbers can be expressed as products of any combination of its factors (except 1 and itself which, whilst true, is not useful).
For example, the number 36.
Its prime factorisation is 22×32 (2 × 2 × 3 × 3)
But it can also be written as a product in several other ways too, such as 4 × 9 or 4 × 3 × 3.
How to avoid this: Make sure you are clear about what a question is asking you to do - does it particularly require prime factorisation? Or are any factors (other than 1 and itself) sufficient to answer the question?
Mistake 3: Not recognising prime numbers
Students that do not recognise prime numbers, can think a value is a composite number and end up wasting time trying to find factors of that value (that don’t exist, apart from 1 and the value itself)
How to avoid this: Learn the prime numbers up to 100. Learning them up to 20 will do most cases, but they get less common after that, and remembering them up to 100 can be useful in harder or more unusual problems.
Frequently asked questions
Is there a pattern to the prime numbers?
No. But up to 20, there is a way to remember them. Starting at 2, 2 is the only even prime number. After that, and up to 20, all odd numbers except 9 and 15 are prime.
So up to 20, we have 2, 3, 5, 7, 11, 13, 17, 19.
My calculator has a prime factorisation feature, can I use that?
Yes, if it is a calculator paper! You may need to be careful about how much working is required to ‘show’ your answer, but a calculator can be used to check your work or help give you something to aim for when stuck.
Do I need to remember the composite numbers?
No. However, by remembering the prime numbers (up to 100) you will automatically know the composite numbers - they are all the other positive whole numbers greater than 1.
How can I quickly find the factors of a composite number?
Firstly, by being very familiar with the times tables up to 12 × 12.
Secondly, by knowing the shortcuts for testing divisibility; any even number is divisible by 2; any number whose digits add up to 3 is divisible by 3; multiples of 5 end in 5 or 0, etc.
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