Algebraic proof about integers

A question is this type if and only if it asks to prove a statement about integer properties (divisibility, even/odd, multiples) using algebraic manipulation, excluding proof by contradiction.

7 questions · Moderate -0.7

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Edexcel P2 2021 June Q3
5 marks Moderate -0.3
3. (i) Prove that for all single digit prime numbers, \(p\), $$p ^ { 3 } + p \text { is a multiple of } 10$$ (ii) Show, using algebra, that for \(n \in \mathbb { N }\) $$( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is not a multiple of } 3$$
Edexcel AS Paper 1 2023 June Q17
5 marks Standard +0.3
  1. In this question \(p\) and \(q\) are positive integers with \(q > p\)
Statement 1: \(q ^ { 3 } - p ^ { 3 }\) is never a multiple of 5
  1. Show, by means of a counter example, that Statement 1 is not true. Statement 2: When \(p\) and \(q\) are consecutive even integers \(q ^ { 3 } - p ^ { 3 }\) is a multiple of 8
  2. Prove, using algebra, that Statement 2 is true.
Edexcel AS Paper 1 2024 June Q14
4 marks Easy -1.8
  1. Prove, using algebra, that
$$n ^ { 2 } + 5 n$$ is even for all \(n \in \mathbb { N }\)
Edexcel AS Paper 1 2021 November Q10
5 marks Moderate -0.3
  1. A student is investigating the following statement about natural numbers.
\begin{displayquote} " \(n ^ { 3 } - n\) is a multiple of 4 "
  1. Prove, using algebra, that the statement is true for all odd numbers.
  2. Use a counterexample to show that the statement is not always true. \end{displayquote}
Edexcel PMT Mocks Q11
4 marks Standard +0.3
11. Prove, using algebra that $$n ^ { 2 } + 1$$ is not divisible by 4 .
Edexcel Paper 1 2023 June Q14
4 marks Easy -1.8
  1. Prove, using algebra, that
$$( n + 1 ) ^ { 3 } - n ^ { 3 }$$ is odd for all \(n \in \mathbb { N }\)
Edexcel Paper 2 2022 June Q11
4 marks Easy -1.2
  1. Prove, using algebra, that
$$n \left( n ^ { 2 } + 5 \right)$$ is even for all \(n \in \mathbb { N }\).