GCSE · Algebraic proof

Proving divisibility resultsExam-style questions

The expand–collect–factorise engine: to prove "always a multiple of k", exhibit the expression as k × (whole number) — and say so.

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13 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Prove that the difference between the squares of any two consecutive even numbers is a multiple of 4.
Level 5 · 3 marksTry it
2
Prove that $n^3 - n$ is a multiple of 6 for every integer $n$ .
Level 5 · 4 marksTry it
3
Prove that the sum of any four consecutive integers is never a multiple of 4.
Level 5 · 3 marksTry it
4
Prove that $n^2 + 3n$ is even for every integer $n$ .
Level 5 · 3 marksTry it
5
Prove that $n^3 + 3n^2 + 2n$ is a multiple of 6 for every integer $n$ .
Level 5 · 4 marksTry it