Divisibility proof for all integers

A question is this type if and only if it asks to prove that an expression involving n (e.g., n³ - n, n² + 2n) is always divisible by a specific number or is always even/odd for all integers n.

19 questions · Moderate -0.1

1.01a Proof: structure of mathematical proof and logical steps
Sort by: Default | Easiest first | Hardest first
OCR MEI C1 2008 January Q9
5 marks Moderate -0.3
9
  1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
  2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
OCR MEI C3 Q1
4 marks Standard +0.5
1 Prove that the product of any three consecutive integers is a multiple of 6 .
OCR MEI C3 2013 June Q2
4 marks Moderate -0.5
2
  1. Factorise fully \(n ^ { 3 } - n\).
  2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .
OCR MEI AS Paper 1 2024 June Q3
3 marks Standard +0.3
3 Prove that, when \(n\) is an even number, \(n ^ { 3 } + 4\) is a multiple of 4 but not a multiple of 8 .
OCR H240/02 2018 September Q4
4 marks Moderate -0.5
4 Prove that the sum of the squares of any two consecutive integers is of the form \(4 k + 1\), where \(k\) is an integer.
AQA AS Paper 2 2021 June Q9
4 marks Standard +0.3
9
  1. Express \(n ^ { 3 } - n\) as a product of three factors. 9
  2. Given that \(n\) is a positive integer, prove that \(n ^ { 3 } - n\) is a multiple of 6
OCR MEI C1 2009 June Q6
3 marks Moderate -0.8
Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]
OCR MEI C1 2011 June Q10
3 marks Moderate -0.8
Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]
OCR MEI C1 2012 June Q9
4 marks Moderate -0.8
Simplify \((n + 3)^2 - n^2\). Hence explain why, when \(n\) is an integer, \((n + 3)^2 - n^2\) is never an even number. Given also that \((n + 3)^2 - n^2\) is divisible by \(9\), what can you say about \(n\)? [4]
OCR MEI C1 2013 June Q9
4 marks Moderate -0.8
\(n - 1\), \(n\) and \(n + 1\) are any three consecutive integers.
  1. Show that the sum of these integers is always divisible by 3. [1]
  2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3. [3]
OCR MEI C1 Q8
3 marks Moderate -0.5
Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]
OCR MEI C1 Q4
5 marks Moderate -0.3
  1. Prove that 12 is a factor of \(3n^2 + 6n\) for all even positive integers \(n\). [3]
  2. Determine whether 12 is a factor of \(3n^2 + 6n\) for all positive integers \(n\). [2]
OCR MEI C3 2011 June Q7
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]
AQA AS Paper 1 2022 June Q9
5 marks Standard +0.3
Integers \(m\) and \(n\) are both odd. Prove that \(m^2 + n^2\) is a multiple of 2 but not a multiple of 4 [5 marks]
AQA AS Paper 2 2023 June Q8
5 marks Standard +0.3
Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4. [5 marks]
AQA AS Paper 2 Specimen Q12
4 marks Moderate -0.3
  1. Given that \(n\) is an even number, prove that \(9n^2 + 6n\) has a factor of 12 [3 marks]
  2. Determine if \(9n^2 + 6n\) has a factor of 12 for any integer \(n\). [1 mark]
OCR MEI AS Paper 2 2018 June Q3
3 marks Moderate -0.8
\(P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\). Prove that \(P^2 - Q^2\) is a multiple of 8. [3]
OCR H240/02 2018 December Q7
5 marks Standard +0.8
  1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
  2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]
Pre-U Pre-U 9795/1 2013 November Q4
4 marks Standard +0.8
Let \(f(n) = 2(5^{n-1} + 1)\) for integers \(n = 1, 2, 3, \ldots\).
  1. Prove that, if \(f(n)\) is divisible by 8, then \(f(n + 1)\) is also divisible by 8. [3]
  2. Explain why this result does not imply that the statement '\(f(n)\) is divisible by 8 for all positive integers \(n\)' follows by mathematical induction. [1]