Proof and Counter-Examples –

Proof and counter-examples are Higher tier GCSE Maths topics that test your ability to reason logically. You must show that a statement is always true (proof) or find a single case where it fails (counter-example).

What Are Proof and Counter-Examples?

An algebraic proof demonstrates that a mathematical statement is true for all cases by using algebra rather than trying individual numbers. For example, proving that the sum of two consecutive numbers is always odd requires you to write the numbers as n and n + 1, add them to get 2n + 1, and explain why this is always odd.

A counter-example is a single specific case that shows a statement is false. For instance, the claim "all prime numbers are odd" is disproved by the counter-example 2, which is prime but even. You only need one counter-example to disprove a statement.

Common proof questions at GCSE involve properties of even and odd numbers, consecutive numbers, and divisibility. You need to know the standard algebraic representations: an even number is 2n, an odd number is 2n + 1, and consecutive numbers are n, n + 1, n + 2, and so on.

Key Formulas

Step-by-Step Method

1. For a proof: represent the numbers algebraically (e.g. 2n for even, 2n + 1 for odd).
2. Perform the required operation (add, subtract, multiply, square, etc.).
3. Simplify and factorise the result to show it matches the required form.
4. Write a concluding statement explaining why the result proves the claim.
5. For a counter-example: find one specific value that makes the statement false and show the calculation.

Worked Example 1 — Foundation Level

Question: This is a Higher only topic. Here is an accessible entry. Show that the statement "the sum of two odd numbers is odd" is false. Give a counter-example.

Working:

Step 1 — Choose two odd numbers: 3 and 5.

Step 2 — Add them: 3 + 5 = 8.

Step 3 — 8 is even, not odd.

Answer: Counter-example: 3 + 5 = 8, which is even. The statement is false.

Worked Example 2 — Higher Level

Question: Prove that the sum of any three consecutive integers is always divisible by 3.

Working:

Step 1 — Let the three consecutive integers be n, n + 1, and n + 2.

Step 2 — Add them: n + (n + 1) + (n + 2) = 3n + 3.

Step 3 — Factorise: 3n + 3 = 3(n + 1).

Step 4 — Since 3(n + 1) is 3 multiplied by an integer, it is always divisible by 3.

Answer: The sum is 3(n + 1), which is always a multiple of 3. QED.

Worked Example 3 — Exam Style

Question: Prove algebraically that the difference between the squares of two consecutive odd numbers is always a multiple of 8. (4 marks)

Working:

Step 1 — Let the two consecutive odd numbers be (2n + 1) and (2n + 3).

Step 2 — Square both: (2n + 3)² = 4n² + 12n + 9 and (2n + 1)² = 4n² + 4n + 1.

Step 3 — Subtract: (4n² + 12n + 9) - (4n² + 4n + 1) = 8n + 8.

Step 4 — Factorise: 8n + 8 = 8(n + 1).

Step 5 — Since 8(n + 1) is 8 multiplied by an integer, the difference is always a multiple of 8.

Answer: The difference is 8(n + 1), which is always divisible by 8. QED.

Common Mistakes

• Using specific numbers instead of algebra for a proof. Showing that 3 + 5 = 8 is even proves nothing in general — you must use 2n and 2m to prove it for all even numbers.
• Not writing a conclusion. After simplifying, you must state why the expression proves the claim (e.g. "this is 2 times an integer, so it is even").
• Thinking one example proves a statement. One example supports it, but only algebra proves it for all cases.

Exam Tips

• For "prove" questions, always use algebra. Start by defining your variables (e.g. "let n be an integer").
• For "show that... is false" or "give a counter-example," one numerical example is enough.
• Memorise: even = 2n, odd = 2n + 1, consecutive = n, n + 1. These are the building blocks of almost every proof at GCSE.

Practice Questions

Q1 (Higher): Prove that the sum of two even numbers is always even.

Q2 (Higher): Give a counter-example to disprove: "if n is a positive integer, then n² + n + 1 is always prime."

Q3 (Higher): Prove that the product of two consecutive integers is always even.

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Related Topics

• Algebraic Proof
• Simplifying Expressions
• Expanding Brackets

Summary

• A proof uses algebra to show a statement is true for all cases, not just some.
• A counter-example is one specific case that disproves a statement.
• Represent even numbers as 2n and odd numbers as 2n + 1.
• After simplifying, always write a concluding sentence explaining why the result proves the claim.
• You only need one counter-example to disprove a statement, but you need algebra to prove one.