Proof involving squares and modular forms

A question is this type if and only if it asks to prove that squares of integers have specific modular properties (e.g., n² is always 0 or 1 mod 3, or n² + 2 is never divisible by 4).

6 questions · Standard +0.4

1.01d Proof by contradiction
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Edexcel P2 2018 Specimen Q4
4 marks Moderate -0.8
Given \(n \in \mathbb { N }\), prove, by exhaustion, that \(n ^ { 2 } + 2\) is not divisible by 4 . \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-12_2658_1943_111_118}
Edexcel P4 2022 June Q9
4 marks Standard +0.3
  1. Use proof by contradiction to show that, when \(n\) is an integer,
$$n ^ { 2 } - 2$$ is never divisible by 4
Edexcel AS Paper 1 2019 June Q15
4 marks Standard +0.8
  1. Given \(n \in \mathbb { N }\), prove that \(n ^ { 3 } + 2\) is not divisible by 8
Edexcel Paper 1 2019 June Q10
6 marks Standard +0.3
  1. (i) Prove that for all \(n \in \mathbb { N } , n ^ { 2 } + 2\) is not divisible by 4
    (ii) "Given \(x \in \mathbb { R }\), the value of \(| 3 x - 28 |\) is greater than or equal to the value of ( \(x - 9\) )." State, giving a reason, if the above statement is always true, sometimes true or never true.
    (2)
AQA AS Paper 1 2018 June Q7
5 marks Standard +0.8
Prove that \(n\) is a prime number greater than \(5 \Rightarrow n^4\) has final digit \(1\) [5 marks]
OCR PURE Q5
5 marks Standard +0.8
\(N\) is an integer that is not divisible by 3. Prove that \(N^2\) is of the form \(3p + 1\), where \(p\) is an integer. [5]