Pure: Proof
Mathematical proof is a logical, rigorous argument that shows a mathematical statement is true for every case it covers, using algebra and accepted facts rather than a handful of examples. A Level Maths requires four methods: proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction, examined throughout the Pure papers.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Identify which proof method the question needs: deduction (direct algebra), exhaustion (check every case in a finite set), counter-example (find one case that fails), or contradiction (assume the opposite and find a contradiction).
- For deduction, represent the numbers algebraically (e.g. 2n for even, 2n+1 for odd) and manipulate the expression until the required property is obvious.
- For exhaustion, list every case in the finite set and verify the statement holds in each one, with no case skipped.
- For counter-example, test small or unusual values (0, negative numbers, non-integers) until one breaks the statement, then state clearly why it fails.
- For contradiction, assume the negation of the statement is true, follow logical steps to reach an impossible conclusion, then conclude the original statement must be true.
- Always finish with a clear concluding sentence linking the algebra back to the original claim.
Worked example
Prove that the sum of any two even integers is always even.
- Let the two even integers be 2m and 2n, where m and n are integers.
- Form the sum: 2m + 2n.
- Factorise: 2m + 2n = 2(m + n).
- Since m + n is an integer, 2(m + n) is a multiple of 2, so it is even.
- Final answer: the sum of any two even integers is always even (proved).
Practice questions
Try each question, then tap to reveal the answer.
Q1Prove that the difference between any two odd integers is always even.Show answer
Answer: Difference = (2m+1) - (2n+1) = 2m - 2n = 2(m-n), a multiple of 2, so always even.
Q2Prove that the square of any even integer is always even.Show answer
Answer: (2n)^2 = 4n^2 = 2(2n^2), a multiple of 2, so always even.
Q3Find a counter-example to disprove the statement: for every positive integer n, 2^n > n^3.Show answer
Answer: n = 9: 2^9 = 512, 9^3 = 729, and 512 < 729, so the statement is false.
Q4Prove by contradiction that there is no smallest positive real number.Show answer
Answer: Assume q is the smallest positive real number; q/2 is also positive and q/2 < q, contradicting q being smallest, so no such number exists.
Q5Prove that if n^2 is divisible by 3 (n an integer), then n is divisible by 3.Show answer
Answer: If n = 3k+1 or 3k+2, n^2 leaves remainder 1 when divided by 3 in both cases, so by the contrapositive n must be divisible by 3 whenever n^2 is.
Q6Prove by contradiction that sqrt(3) is irrational.Show answer
Answer: Assume sqrt(3) = a/b in lowest terms; then a^2 = 3b^2 forces a and then b to both be divisible by 3, contradicting lowest terms, so sqrt(3) is irrational.
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
Prove that the product of any two consecutive even integers is divisible by 8.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Disprove the statement: for every positive integer n, n^2 + n + 41 is prime.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 3 available
Prove by contradiction that there are no two positive integers x and y for which x^2 - y^2 = 1.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 6 available
See real past-paper questions on pure: proof, organised by topic with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the pure: proof worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Other cuts of this worksheet:
Next topics
Not quite what you needed?
Tell us what is missing on pure: proof, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.