Prime numbers come up in every maths exam — from KS2 right through to GCSE and A Level. This guide explains exactly what a prime number is, what makes a number prime, how to check if any number is prime, and gives you a complete list of all prime numbers up to 100.
Let’s make that even simpler before we go further. A factor is just a number that divides into another number without leaving a remainder. So when we say a prime number has exactly two factors, we mean the only numbers that divide into it exactly are 1 and the number itself.
For example — take the number 7. What divides into 7 exactly? Only 1 and 7. Nothing else. So 7 is a prime number.
Now take the number 6. What divides into 6? The numbers 1, 2, 3, and 6 all divide into it. That is four factors — too many. So 6 is not prime.
No — and this is one of the most common wrong answers in GCSE maths. 1 is not a prime number.
Here is why. The definition says a prime number must have exactly two factors. The number 1 only has one factor — itself. It does not have two different factors, so it does not fit the definition. It sits in its own special category: neither prime nor composite.
⚠️ Exam watch: Examiners love testing whether students include 1 in lists of prime numbers. It is not prime. Do not write it down when you are asked to list prime numbers.
In the grid below, every dark blue number is prime. The grey ones are composite (not prime). The number 1 is shown separately because it is neither.
If you count the dark blue boxes, you will find there are 15 prime numbers between 1 and 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
There are exactly 25 prime numbers between 1 and 100. Here they are organised by range — a format that is very useful for revision because exam questions often ask you to find primes “between” two values:
✅ Memory trick for the first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Say them out loud five times. They come up so often in GCSE maths questions that knowing them without thinking saves you real time in the exam.
Every whole number greater than 1 is either prime or composite. There is no middle ground.
A composite number is a number with more than two factors. In simple terms — it can be divided exactly by at least one number other than 1 and itself. For example, 15 is composite because 3 and 5 also divide into it.
| Type | Definition | Factors | Example |
|---|---|---|---|
| Prime | Exactly 2 factors | 1 and itself only | 7 → factors: 1, 7 |
| Composite | More than 2 factors | 1, itself, and others | 12 → factors: 1, 2, 3, 4, 6, 12 |
| 1 | Special — neither | Only 1 factor | 1 → factors: 1 |
📘 What is a non prime number? Any whole number greater than 1 that is not prime is called a composite number. So 4, 6, 8, 9, 10, 12… are all non-prime (composite). The number 1 is also non-prime but sits in its own special category.
Yes — exactly one. 2 is the only even prime number, and it is one of the most important facts in GCSE maths.
Here is the reasoning. Every even number can be divided by 2. So every even number — other than 2 itself — has at least three factors: 1, 2, and the number itself. That means it cannot be prime. The number 2 is the exception because when you divide 2 by 2, you get 2 itself — so it still only has two factors.
🔑 Remember this: 2 is the only even prime number. Every other prime number is odd. But not every odd number is prime — for example, 9 is odd but it factors as 1 × 9 and 3 × 3, so it has three factors and is not prime.
When an exam question asks you to check whether a specific number is prime, here is the method that works every time. You do not need to test every number — you only need to test prime numbers up to the square root of your number.
For example, to check whether 37 is prime:
Square root of 97 ≈ 9.8 — so we test primes up to 9: that is 2, 3, 5, and 7
97 ÷ 2 = 48.5 → not exact. 97 is odd, so 2 does not divide in.
9 + 7 = 16 → 16 is not a multiple of 3. So 3 does not divide in.
97 does not end in 0 or 5. So 5 does not divide in.
97 ÷ 7 = 13.857… → not exact. So 7 does not divide in.
Square root of 91 ≈ 9.5 — so we test primes 2, 3, 5, and 7
91 is odd → 2 does not divide in
9 + 1 = 10 → not a multiple of 3
Does not end in 0 or 5
91 ÷ 7 = 13 exactly ✓ — 7 divides in perfectly!
⚠️ 91 is the classic trick question. Many students think 91 is prime because it does not look like it has small factors. But 91 = 7 × 13. If you get asked this in an exam, remember it!
Every composite number can be written as a product of prime numbers. This is called its prime factorisation. It is a key topic in GCSE maths and comes up in questions about HCF (highest common factor) and LCM (lowest common multiple).
The idea is simple: keep dividing a number by its smallest prime factor until you reach 1.
✅ Use a factor tree! A factor tree is the easiest way to find the prime factorisation of a number. Start by finding any two factors that multiply to give your number, then keep splitting each factor until every branch ends in a prime number.
📘 Struggling with prime factorisation, HCF, or LCM? Our GCSE maths tutors cover every Number topic in the specification, with real past paper questions and examiner-level feedback.
Book Free SessionIf you are in primary school (Year 5 or Year 6) and you need to understand what a prime number is, here is the simplest way to think about it:
A prime number is a number you cannot make into a rectangle.
Think of it like arranging counters or tiles. If a number is composite, you can arrange that many tiles into a neat rectangle in more than one way. For example, 12 tiles can be arranged as 2 × 6, 3 × 4, or 1 × 12.
But if a number is prime, the only rectangle you can make is a single long row — 1 × the number. You cannot split it any other way. That is because it has no factors except 1 and itself.
✅ KS2 prime numbers to know: For Year 5 and Year 6, the most important prime numbers are 2, 3, 5, 7, 11, and 13. Being able to recognise these and explain why they are prime is usually enough at KS2 level.
Two prime numbers that are only 2 apart from each other are called twin primes. Examples: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43). Twin primes are a fascinating area of maths — mathematicians still do not know if there are infinitely many pairs of them.
Two numbers are co-prime (or coprime) when their highest common factor is 1. They do not have to be prime themselves — they just cannot share any common factors apart from 1. For example, 8 and 9 are co-prime because nothing other than 1 divides into both of them.
A super prime is a prime number whose position in the list of primes is also prime. The 2nd prime is 3 (prime position → 3 is super prime). The 3rd prime is 5 (prime position → 5 is super prime). The 5th prime is 11 (prime position → 11 is super prime). The sequence starts: 3, 5, 11, 17, 31, 41, 59, 67, 83…
The opposite of a prime number is a composite number — a number with more than two factors. 4, 6, 8, 9, 10, 12, 14, 15 are all composite numbers.
Prime numbers are not just a maths classroom topic. They are genuinely important in the real world — and one field in particular relies on them completely.
Every time you buy something online or log into an app, your data is protected by encryption that relies on prime numbers. It works because multiplying two huge primes together is easy, but factorising the result back into those two primes is incredibly hard — even for the most powerful computers.
The RSA encryption system — used by virtually every bank, government, and tech company — is built entirely on the mathematical difficulty of prime factorisation. Your credit card details are safe because of prime numbers.
Some composers deliberately use prime-numbered time signatures (like 5/4 or 7/8) because they cannot be broken into smaller equal units, creating rhythms that feel unusual and interesting to the human ear.
Certain cicadas emerge from the ground after exactly 13 or 17 years — both prime numbers. Scientists think this is because prime cycles minimise the chance of the cicadas coinciding with predator population peaks.
Our examiner-qualified GCSE maths tutors cover every topic in the specification — from prime numbers and factors to algebra, geometry, and beyond. Book a free consultation today.
Book Your Free Consultation →A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. The simplest way to remember it: if you can divide a number exactly by anything other than 1 and itself, it is NOT prime. Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23.
In maths, a prime number is a natural number with exactly two distinct factors. Prime numbers are the building blocks of all other numbers — every whole number can be written as a product of prime numbers (prime factorisation). This underpins topics like HCF, LCM, and is the foundation of modern encryption and internet security.
No. 1 is not a prime number. A prime must have exactly two factors, but 1 only has one factor — itself. The number 1 is neither prime nor composite. This is one of the most commonly tested facts in GCSE maths, so remember it.
2 is the only even prime number. Every other even number can be divided by 2 (giving it at least three factors), so no other even number can be prime.
A composite number is a whole number greater than 1 that has more than two factors. For example, 12 is composite because it can be divided by 1, 2, 3, 4, 6, and 12. Every whole number greater than 1 is either prime (exactly 2 factors) or composite (more than 2 factors).
To check if a number is prime, divide it by every prime number up to its square root. If none of them divide exactly, the number is prime. For example, to check 53: the square root of 53 is about 7.3, so you test 2, 3, 5, and 7. None divide 53 exactly, so 53 is prime.
The prime numbers between 1 and 10 are: 2, 3, 5, 7. That is four prime numbers. Note that 1 is not prime, and neither are 4, 6, 8, 9, or 10.
No — 91 is NOT a prime number. It equals 7 × 13, so it has factors other than 1 and 91. This is a very common trick question in GCSE maths because 91 looks prime but it is not.
Writing a number as a product of prime numbers means expressing it through prime factorisation. For example, 60 = 2 × 2 × 3 × 5. Every composite number can be written this way, and the factorisation is always unique (this is called the Fundamental Theorem of Arithmetic). It is an important skill for GCSE maths, used in finding HCF and LCM.
For KS2 students, a prime number is a number that cannot be divided equally into smaller groups except as 1 group or individual items. The key prime numbers to know at KS2 are 2, 3, 5, 7, 11, and 13. If you can arrange the tiles/counters into more than one rectangle shape, the number is not prime.
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