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GCSE Maths · Algebra
Algebraic proof GCSE Questions and Worked Answers
An algebraic proof represents every allowed case with letters, uses valid algebra and ends by explaining why the resulting form establishes the claim. A few numerical examples cannot prove an unrestricted claim.
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Start with the meaning
What you need to know about algebraic proof
If counters can be put into pairs with none left over, their number is even. Two piles made entirely of pairs still make a pile of pairs when combined. That explains why adding two even counts gives an even count without trying every possible pile.
See the idea first
Letters let the argument cover every allowed number
An integer is a whole number, including zero and negative whole numbers. If n is any integer, 2n represents an even integer and 2n + 1 an odd integer. Use different letters for two arbitrary integers: they need not be equal.
| Numbers described | Representation | Reason |
|---|---|---|
| Any even integer | 2n | a multiple of 2 |
| Any odd integer | 2n + 1 | one more than a multiple of 2 |
| Consecutive integers | n, n + 1 | the next integer is one larger |
| Consecutive odd integers | 2n + 1, 2n + 3 | odd integers are two apart |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Prove an even or odd result
What the problem asks: Prove that the sum of two odd integers is even.
How to solve it: Write them as 2a + 1 and 2b + 1, with a and b integers. Their sum is 2(a + b + 1), twice an integer, so it is even.
Prove divisibility
What the problem asks: Prove that the sum of three consecutive integers is divisible by 3.
How to solve it: Write n, n + 1, n + 2. Their sum is 3n + 3 = 3(n + 1). Since n + 1 is an integer, the sum is a multiple of 3.
Disprove a universal claim
What the problem asks: Disprove ‘every prime number is odd’.
How to solve it: The number 2 is prime and even. One allowed counterexample is enough because the claim said every prime.
A reliable routine
Build an argument, not a fixed calculation
There is no universal algorithm for every proof. For claims about integers, a useful approach is to encode the stated conditions and transform the expression into a form that makes the required property visible.
- State what the letters represent and their allowed values.
- Write the full expression described in the claim.
- Expand, collect or factorise using equal expressions; do not assume the conclusion.
- Translate the final form into words: for example, ‘three times an integer, so divisible by three’.
Check: A counterexample disproves a universal claim. A supporting example does not prove one unless every allowed case has been checked.
Fully worked
Algebraic proof GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Two even integers
Question
Prove that the sum of any two even integers is even.
Let a and b be integers, so the even numbers are 2a and 2b.
Since a + b is an integer, this is twice an integer. Therefore the sum is even. Tip: use separate letters so the two even integers can be chosen independently; they are not forced to be equal.
Example 2
Two odd integers
Question
Prove that the sum of any two odd integers is even.
Write the odd integers as 2a + 1 and 2b + 1.
The bracket is an integer, so the sum is even.
Example 3
Three consecutive integers
Question
Prove that three consecutive integers have a sum divisible by 3.
Let the first integer be n.
This is three times the integer n + 1, proving divisibility by 3.
Example 4
Square of an odd integer
Question
Prove that the square of any odd integer is odd.
Write the integer as 2n + 1.
The bracket is an integer. Twice an integer plus one is odd.
Example 5
Consecutive squares
Question
Prove that the difference between the squares of two consecutive integers is odd.
Subtract the square of n from that of n + 1.
This has the form of an odd integer. The sign of the difference does not affect its oddness.
Example 6
A counterexample
Question
Disprove: ‘If an integer is divisible by 3, it is divisible by 6.’
Choose 9. It is divisible by 3 because 9 ÷ 3 = 3, but not by 6 because 9 ÷ 6 = 1.5 is not an integer. This one counterexample disproves the claim.
10 original questions · total 27 marks
Algebraic proof GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 32 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Represent a multiple
Write an expression for any multiple of 5 and define your letter.
Show worked answer
Use 5n, where n is an integer. This covers all integer multiples, not only positive ones.
Consecutive even integers
Write two consecutive even integers using one letter.
Show worked answer
Use 2n and 2n + 2, where n is an integer. Consecutive even integers differ by 2, not 1.
Odd plus even
Prove that an odd integer plus an even integer is odd.
Show worked answer
Let the numbers be 2a + 1 and 2b.
As a + b is an integer, the result is odd.
Odd times odd
Prove that the product of two odd integers is odd.
Show worked answer
Let a, b be integers.
The bracket is an integer, so the result is odd.
Two consecutive integers
Prove that the sum of two consecutive integers is odd.
Show worked answer
Let them be n and n + 1.
Since n is an integer, the sum is odd.
A difference divisible by four
Prove that the difference of the squares of two consecutive odd integers is divisible by 4.
Show worked answer
Use 2n + 1 and 2n + 3.
The bracket is an integer, so the difference is divisible by 4 (in fact, by 8).
Consecutive product
Prove that the product of two consecutive integers is even.
Show worked answer
Write the product n(n + 1). Of consecutive integers, one is even: if n is odd, n + 1 is even; if n is even, it already supplies a factor of 2. The product is therefore even.
Expand an identity
Prove that for every real x.
Show worked answer
Expand both squares, keeping the subtraction bracket.
Every step is an identity, so it holds for all real x.
Disprove a square claim
Disprove: ‘Squaring any positive number makes it bigger.’
Show worked answer
Take . Then . This positive input gives a smaller square, so the universal claim is false.
Finish the reasoning
A proof ends with , where k is an integer. Explain why the result is divisible by 7.
Show worked answer
Since k is an integer, k + 2 is an integer too. The expression is 7 times an integer, so it is a multiple of 7. Merely showing a factor of 7 without the integer condition would not justify integer divisibility.
Examiner-style feedback
Common algebraic proof mistakes
Examples support a conjecture but cannot prove every integer case.
Consecutive odd numbers are two apart: 2n + 1 and 2n + 3.
Explain why the final form establishes the required property.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Define letters and conditions.
- Represent the full claim.
- Use valid equalities.
- State the conclusion from the final form.
Quick answers
Algebraic proof FAQ
Must a proof be long?
No. A short argument is enough if it covers every allowed case and explains the conclusion.
Can a proof use words?
Yes. Words are essential for definitions, cases and explaining what the algebra proves.
Content standards
Curriculum and rights review
A6 formal algebraic proofs and counterexamples: Higher tier, with explicit integer and real-number domains. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references