Number
Grade 1-3Types of Number
Before anything else in GCSE Maths, you need to be fluent in the different types of number: square, cube, prime and triangular. They show up as standalone questions on every paper, and they quietly sit underneath topics like factorising, surds and sequences. This lesson covers all four, with a full practice paper and mark scheme at the end.
What you need to know
- Integers are whole numbers; a prime has exactly two factors, 1 and itself, so 1 is not prime.
- The first primes are 2, 3, 5, 7, 11, 13, and 2 is the only even prime.
- Square numbers come from n squared (1, 4, 9, 16); cube numbers from n cubed (1, 8, 27, 64).
- A rational number can be written as a fraction; irrational numbers like pi and root 2 cannot.
Written by Asad, co-founder of Teachably
Square and cube numbers
A square number is what you get when you multiply a whole number by itself. The first ten are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Written with power notation, 7² means 7 × 7 = 49.
A cube number is what you get when you multiply a whole number by itself three times. The first five are 1, 8, 27, 64, 125. Written with power notation, 4³ means 4 × 4 × 4 = 64, not 4 × 3.
Prime numbers
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are the first ten primes.
1 is not a prime number, because it only has one factor, not two. 2 is the only even prime number, because every other even number can be divided by 2, giving it a third factor.
To check whether a larger number is prime, test whether it divides exactly by the small primes in turn (2, 3, 5, 7, 11...) up to its square root. If none of them divide in exactly, the number is prime.
Triangular numbers
Triangular numbers come from stacking dots into a triangle: 1, then 1+2, then 1+2+3, and so on. The sequence begins 1, 3, 6, 10, 15, 21, 28.
The nth triangular number can be found with the formula n(n + 1) / 2, without having to add up the whole sequence. For example, the 10th triangular number is (10 × 11) / 2 = 55.
Worked examples
Three exam-style questions, fully solved
Write down the value of 7².
Easy- 1.7² means 7 multiplied by itself: 7 × 7
- 2.Calculate: 7 × 7 = 49
Answer: 49
Here is a list of numbers: 8, 9, 11, 14. (a) Write down the square number from the list. (b) Write down the prime number from the list.
Medium- 1.Check each number in turn: 8 = 2 × 2 × 2 (a cube number, not square), 9 = 3 × 3 (a square number), 11 has only the factors 1 and 11 (a prime number), 14 = 2 × 7 (neither square nor prime)
- 2.The square number is 9
- 3.The prime number is 11
Answer: (a) 9, (b) 11
Is 51 a prime number? Give a reason for your answer.
Hard- 1.Test whether 51 divides exactly by small primes: 51 is not even, so it is not divisible by 2
- 2.Check 3: 51 ÷ 3 = 17, which is a whole number
- 3.So 51 = 3 × 17, meaning it has factors other than 1 and itself
Answer: No, 51 is not prime, because 51 = 3 × 17
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Work out 9².
What is the smallest cube number that is greater than 1?
From this list, write down the prime number: 15, 16, 17, 18.
Work out the 6th triangular number.
Work out the difference between 3³ and 5².
How many prime numbers are there between 20 and 30?
Avoid these
The mistakes students make most often
Thinking 1 is a prime number. It only has one factor, so it does not meet the definition of prime.
Working out 4³ as 4 × 3 = 12 instead of 4 × 4 × 4 = 64. The small number tells you how many times to multiply, not what to multiply by.
Confusing square numbers with even numbers, when a square number can be odd or even (9 and 25 are both odd square numbers).
Using the triangular number formula but forgetting to divide by 2, giving n(n + 1) instead of n(n + 1) / 2.
On a "give a reason" question, writing only "yes" or "no" without showing a factor pair or working, which loses the mark even with the correct answer.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
Teachably is a GCSE Maths tutoring practice
This hub is our free resource and it stays free. If you would like one-to-one help alongside it, a short call is the best place to start. There is also a free video course for parents.