Sheet № 205 · Higher only · AQA · Edexcel · OCR
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).
§Key definitions
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.
Answer:
Counter-example: 3 + 5 = 8, which is even. The statement is false.
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.
§Formulas to memorise
Even number = 2n, Odd number = 2n + 1 (where n is any integer)
Consecutive integers: n, n + 1, n + 2, ...
Consecutive even numbers: 2n, 2n + 2, 2n + 4, ...
Worked example
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:
⚠ 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.