Proof by exhaustion

A question is this type if and only if it explicitly requires checking all possible cases from a finite set to prove a statement.

2 questions · Moderate -0.8

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Edexcel Paper 1 2021 October Q15
6 marks Moderate -0.8
  1. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\)
$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.
AQA Paper 2 2022 June Q6
5 marks Moderate -0.8
6
  1. Asif notices that \(24 ^ { 2 } = 576\) and \(2 + 4 = 6\) gives the last digit of 576 He checks two more examples: $$\begin{array} { l c } 27 ^ { 2 } = 729 & 29 ^ { 2 } = 841 \\ 2 + 7 = 9 & 2 + 9 = 11 \\ \text { Last digit } 9 & \text { Last digit } 1 \end{array}$$ Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif's conclusion is not correct. 6
  2. Claire tells Asif that he should look only at the last digit of the number being squared. $$\begin{array} { c c } 27 ^ { 2 } = 729 & 24 ^ { 2 } = 576 \\ 7 ^ { 2 } = 49 & 4 ^ { 2 } = 16 \\ \text { Last digit } 9 & \text { Last digit } 6 \end{array}$$ Using Claire's method determine the last digit of \(23456789 { } ^ { 2 }\)
    [0pt] [1 mark] 6
  3. Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8