Proof by exhaustion with cases

A question is this type if and only if it requires proving a statement by checking all possible cases systematically, often involving modular arithmetic (e.g., n = 2k or n = 2k+1, or n ≡ 0, 1, 2 mod 3) or a finite set of values.

8 questions · Moderate -0.6

1.01a Proof: structure of mathematical proof and logical steps
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OCR MEI C3 2007 January Q4
3 marks Moderate -0.8
4 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.
Edexcel Paper 2 2020 October Q16
4 marks Moderate -0.3
  1. Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3
OCR PURE Q5
5 marks Moderate -0.3
5
  1. Prove that the following statement is not true. \(m\) is an odd number greater than \(1 \Rightarrow m ^ { 2 } + 4\) is prime.
  2. By considering separately the case when \(n\) is odd and the case when \(n\) is even, prove that the following statement is true. \(n\) is a positive integer \(\Rightarrow n ^ { 2 } + 1\) is not a multiple of 4 .
OCR H240/02 2022 June Q7
8 marks Moderate -0.3
7 It is given that any integer can be expressed in the form \(3 m + r\), where \(m\) is an integer and \(r\) is 0,1 or 2 . Use this fact to answer the following.
  1. By considering the different values of \(r\), prove that the square of any integer cannot be expressed in the form \(3 n + 2\), where \(n\) is an integer.
  2. Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different. By considering the different values of \(r\), determine the probability that the sum of these three integers is divisible by 3 .
AQA Paper 1 2020 June Q5
2 marks Easy -1.8
5 Prove that, for integer values of \(n\) such that \(0 \leq n < 4\) $$2 ^ { n + 2 } > 3 ^ { n }$$
OCR MEI C3 2012 January Q4
2 marks Standard +0.8
Prove or disprove the following statement: 'No cube of an integer has 2 as its units digit.' [2]
OCR MEI C3 Q11
3 marks Moderate -0.5
Use the method of exhaustion to prove the following result. No 1- or 2-digit perfect square ends in 2, 3, 7 or 8 State a generalisation of this result. [3]
WJEC Unit 1 2024 June Q4
3 marks Easy -1.2
Given that \(n\) is an integer such that \(1 \leqslant n \leqslant 6\), use proof by exhaustion to show that \(n^2 - 2\) is not divisible by 3. [3]