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Prime Numbers

FoundationChallengeAQAEdexcelOCR

Prime Numbers is a key number topic at GCSE Maths. This Foundation worksheet gives you exam-style questions on identifying prime numbers, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. A prime has exactly two factors, 1 and itself — so 1 is not prime.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

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Topic overview

A prime number has exactly two factors: \(1\) and itself. The primes begin \(2, 3, 5, 7, 11, 13, 17, 19, 23\) and \(29\), and knowing them up to \(30\) is worth committing to memory.

Two details trip people up. One is not prime, because it has only a single factor rather than two. And \(2\) is prime — it is the only even prime, since every other even number is divisible by \(2\).

Primes matter because every whole number above \(1\) can be written as a product of primes in exactly one way. That fact underpins finding highest common factors and lowest common multiples.

Revision notes

Testing whether a number is prime

Try dividing by each prime in turn: \(2, 3, 5, 7\) and so on. You only need to test primes up to the square root of the number.

For \(91\), test \(2, 3, 5, 7\). Since \(91 = 7 \times 13\), it is not prime, despite looking like one.

Why 1 is not prime

A prime has exactly two distinct factors. The number \(1\) has only one factor, itself, so it fails the definition.

This is not a technicality — if \(1\) counted as prime, numbers would no longer have a unique prime factorisation.

Useful divisibility checks

A number is divisible by \(2\) if it is even, by \(3\) if its digits add to a multiple of \(3\), by \(5\) if it ends in \(0\) or \(5\), and by \(9\) if its digits add to a multiple of \(9\).

So \(123\) has digits summing to \(6\), which is divisible by \(3\), so \(123\) is not prime.

Key points

  • A prime has exactly two factors: 1 and itself.
  • 1 is not a prime number.
  • 2 is the only even prime.
  • Primes to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
  • Test divisibility by primes up to the square root.
  • Every whole number above 1 has a unique prime factorisation.

Worked examples

Example 1

Is \(51\) a prime number?

Working

\[5 + 1 = 6\]add the digits to test divisibility by 3
\[6 \text{ is a multiple of } 3\]so 51 is divisible by 3
\[51 = 3 \times 17 \text{, not prime}\]state the conclusion with a factor pair

Example 2

Write down all the prime numbers between \(20\) and \(30\).

Working

\[21, 22, 24, 25, 26, 27, 28 \text{ have other factors}\]eliminate the composite numbers
\[23, 29\]these have exactly two factors each

Example 3

Explain why \(2\) is the only even prime number.

Working

\[2 = 1 \times 2\]2 has exactly two factors, so it is prime
\[\text{Every other even number divides by } 2\]so it has at least three factors and cannot be prime

Common mistakes

  • Calling 1 a prime number.

    A prime needs exactly two distinct factors, and 1 has only one.

  • Assuming all odd numbers are prime.

    9, 15, 21 and 25 are all odd but have additional factors.

  • Forgetting 2 is prime.

    It is the only even prime, and it is often wrongly excluded.

  • Testing every number up to the value itself.

    You only need to test primes up to the square root, which saves a great deal of time.

Exam tips

  • Memorise the primes up to 30 — they appear constantly.
  • Use the digit-sum test for 3 and 9 before trying long division.
  • Give a factor pair when explaining why a number is not prime.
  • Only test divisibility by primes up to the square root.

Key terms

Prime number
A number with exactly two factors, 1 and itself.
Composite number
A number with more than two factors.
Divisible
Able to be divided exactly with no remainder.
Prime factorisation
Writing a number as a product of primes.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.