Prove polynomial divisibility property

A question is this type if and only if it asks to prove (by factorization or other means, not necessarily induction) that a polynomial expression is divisible by an integer for all integer n.

6 questions · Standard +0.2

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Edexcel AS Paper 1 2022 June Q14
4 marks Standard +0.3
  1. (i) A student states
    "if \(x ^ { 2 }\) is greater than 9 then \(x\) must be greater than 3 "
Determine whether or not this statement is true, giving a reason for your answer.
(ii) Prove that for all positive integers \(n\), $$n ^ { 3 } + 3 n ^ { 2 } + 2 n$$ is divisible by 6
OCR Further Pure Core 1 2018 March Q3
4 marks Moderate -0.3
3 Prove by mathematical induction that, for all integers \(n \geqslant 1 , n ^ { 5 } - n\) is divisible by 5 .
OCR MEI C3 2016 June Q7
4 marks Standard +0.8
You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]
OCR MEI C3 Q3
4 marks Moderate -0.3
  1. Factorise fully \(n^3 - n\). [2]
  2. Hence prove that, if \(n\) is an integer, \(n^3 - n\) is divisible by 6. [2]
OCR MEI C3 Q6
4 marks Standard +0.3
  1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
  2. Hence prove that if \(n\) is a positive integer then \(3^{2n} - 1\) is divisible by 8. [3]
OCR H240/02 2020 November Q5
5 marks Standard +0.3
  1. Determine the set of values of \(n\) for which \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) are positive integers. [3]
A 'Pythagorean triple' is a set of three positive integers \(a\), \(b\) and \(c\) such that \(a^2 + b^2 = c^2\).
  1. Prove that, for the set of values of \(n\) found in part (a), the numbers \(n\), \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) form a Pythagorean triple. [2]