Definition
A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself.
Explanation
Prime numbers are special numbers in maths. They can only be divided evenly by two numbers: 1 and themselves. Prime numbers are positive whole numbers (also known as positive integers or natural numbers).
Let's look at this more closely. Every prime number has exactly two factors. A factor is a number that divides evenly into another number with no remainder.
Take the number 7 as an example. You can only divide 7 evenly by 1 and by 7. Try dividing 7 by 2, 3, 4, 5, or 6 - you'll always get a non-whole number. This makes 7 a prime number.
The number 1 is not considered a prime number. This might seem confusing because 1 only has one factor (itself), but prime numbers are natural numbers greater than or equal to 2 (and have exactly two distinct (different) factors by definition).
The number 2 is the smallest prime number. It's also the only even prime number. All other even numbers can be divided by 2, as well as 1 and their self, which means they have more than two factors.
Here are the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Numbers that are not prime are called composite numbers. Composite numbers have more than two factors. For example, 6 is composite because it has four factors: 1, 2, 3, and 6.
Prime numbers become less common as numbers get larger. There are 25 prime numbers between 1 and 100, but only 21 prime numbers between 101 and 200.
Prime numbers are important in many areas of maths. They're the building blocks of all other numbers. Every number can be written as a product of prime numbers. For example, the composite number 210 is the product of 2, 3, 5 and 7; 2 x 3 x 5 x 7 = 210.
Example
Question: Which of these numbers are prime: 13, 15, 17, 21?
Solution:
Let's check each number by finding all its factors.
Checking 13:
Factors of 13: 1 and 13
13 ÷ 1 = 13 ✓
13 ÷ 13 = 1 ✓
No other numbers divide evenly into 13
13 has exactly two factors, so 13 is prime
Checking 15:
Let's test: 15 ÷ 3 = 5 ✓
So 3 and 5 are also factors of 15
Factors of 15: 1, 3, 5, 15
15 has four factors, so 15 is not prime (it's composite)
Checking 17:
Test possible factors: 2, 3, 4, 5... up to 16
None of these divide evenly into 17
Factors of 17: 1 and 17
17 has exactly two factors, so 17 is prime
Checking 21:
Let's test: 21 ÷ 3 = 7 ✓
So 3 and 7 are also factors of 21
Factors of 21: 1, 3, 7, 21
21 has four factors, so 21 is not prime (it's composite)
Answer: 13 and 17 are prime numbers. 15 and 21 are not prime.
Common mistakes (and how to avoid them)
Mistake 1: Thinking 1 is a prime number
Many students assume that 1 is prime because it seems to fit the definition. After all, 1 is only divisible by itself.
The reason 1 is not prime is that prime numbers must have exactly two distinct (different) factors. The number 1 only has one factor (itself), so it doesn't qualify.
How to avoid this: Remember that prime numbers need exactly two factors: 1 and the number itself. Since 1 and itself are the same thing for the number 1, it only has one factor.
Mistake 2: Forgetting that 2 is prime
Students sometimes think 2 is not prime because it's even. They assume all even numbers are composite.
All even numbers, except for, 2, are composite. The number 2 has only two distinct factors: 1 and 2.
How to avoid this: Remember that 2 is the only even prime number. All other even numbers can be divided by 2, giving them at least three factors (1, 2, and themselves).
Mistake 3: Not checking all possible factors
When testing if a number is prime, some students stop checking too early or miss certain factors.
For example, when checking if 49 is prime, they might test 2, 3, 5 and then give up. But 49 = 7 × 7, so 7 is a factor they missed.
How to avoid this: When checking if a number is prime, test all numbers up to the square root of that number. For 49, you need to test up to 7 (since 49 = 7).
Mistake 4: Confusing prime and odd numbers
Students sometimes think all odd numbers are prime, or that all prime numbers (except 2) are odd.
While all prime numbers, except 2, are odd, not all odd numbers are prime. For example, 9, 15, 21, 25, 27 are all odd but not prime.
How to avoid this: Don't use shortcuts. Always check the factors properly. Being odd is necessary for most primes, but it's not sufficient on its own.
Mistake 5: Making errors with larger numbers
Students often struggle to identify prime numbers when the numbers get bigger, like those over 50.
They might incorrectly identify numbers like 51, 57, or 63 as prime without proper checking.
How to avoid this: Use systematic checking. For larger numbers, test divisibility by small primes first (2, 3, 5, 7, 11, etc.). This will help you spot composite numbers quickly.
Frequently asked questions
Why isn't 1 considered a prime number?
The number 1 isn't prime because prime numbers must have exactly two factors - those factors have to be different (distinct). All natural numbers are divisible by 1 and their self, so this fits the definition of having two distinct factors. The number 1 has only one factor (itself), it doesn't meet this requirement. Mathematicians decided this definition works best for the way prime numbers are used in advanced maths.
What's the largest prime number?
There's no largest prime number. Mathematicians proved over 2,000 years ago that there are infinitely many prime numbers. However, finding very large prime numbers requires powerful computers and can take years.
Are there any patterns in prime numbers?
Prime numbers don't follow simple patterns, which makes them fascinating to mathematicians. However, there are some interesting observations. For example, except for 2 and 3, all primes are either 1 more or 1 less than a multiple of 6.
How do you quickly check if a number is prime?
For smaller numbers (under 100), you only need to test if the number is divisible by primes up to its square root. For larger numbers, there are computer algorithms, but these are beyond GCSE level.
What's the difference between prime and composite numbers?
Prime numbers have exactly two factors (1 and themselves). Composite numbers have more than two factors. The number 1 is neither prime nor composite.
Related GCSE Mathematics glossaries
Examiner-written GCSE Maths revision resources that improve your grades 2x
- Written by expert teachers and examiners
- Aligned to exam specifications
- Everything you need to know, and nothing you don’t
Share this article
written revision resources that improve your