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Paper 1
Proof & Algebra
Both
Write a clean chain of algebra for proof.
Key Facts
- Odd numbers: 2n+1. Even numbers: 2n.
- Consecutive integers: n and n+1.
- Show factorization to prove divisibility.
- One counter-example disproves a conjecture.
- QED means "proof complete".
Topics Covered
Algebraic Proof
What you need to know
- •Represent odd numbers as 2n+1 and even numbers as 2n.
- •Expand and simplify to show a common factor.
- •Finish by factoring to prove divisibility.
- •Use clear logical steps throughout.
Exam Tips
- State assumptions clearly and show each algebraic step.
- Conclude with a clear statement.
Proof by Counter-example
What you need to know
- •To disprove a statement, find one counter-example.
- •Check the statement carefully before choosing values.
Exam Tips
- One counter-example is enough to disprove.
Parity Arguments
What you need to know
- •Odd + odd = even
- •Even + even = even
- •Odd × odd = odd
- •Even × anything = even
- •Use n and n+1 as consecutive integers.
Exam Tips
- Use parity to show divisibility by 2.
Key Terms
Conjecture
A statement believed to be true and needing proof.
Counter-example
An example that shows a statement is false.
Parity
Whether a number is odd or even.
Divisibility
When a number divides another with no remainder.
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Common Exam Questions
Prove that n(n+1) is always even.
3 markseasyPaper 1
Model Answer
One of n or n+1 is even, so the product is even. QED.
What examiners want to see
- ✓Use parity argument.
- ✓Clear explanation.
Prove that (2n+1)² - 1 is divisible by 8.
4 marksmediumPaper 1
Model Answer
(2n+1)² - 1 = 4n² + 4n + 1 - 1 = 4n² + 4n = 4n(n+1). One of n or n+1 is even, so 4n(n+1) is divisible by 8.
What examiners want to see
- ✓Expand correctly.
- ✓Factor out 4.
- ✓Use parity of consecutive integers.
Disprove: "All prime numbers are odd."
1 markeasyPaper 1
Model Answer
Counter-example: 2 is prime and even.
What examiners want to see
- ✓Give valid counter-example.
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