A-level · Proof

Proof by exhaustionExam-style questions

Proving a statement by splitting it into a finite, complete set of cases and checking every one — either a short list of values, or all integers split by parity or remainder. Why the cases must leave no gaps, and why exhaustion only works when the cases really are finite.

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6 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
(a) By writing $n$ in terms of its remainder on division by $3$ , prove that, for every integer $n$ , the remainder when $n^3$ is divided by $9$ is $0$ , $1$ or $8$ .

(4 marks)
(b) The integers $x$ , $y$ and $z$ are each chosen at random from the set $\{1, 2, 3, \ldots, 90\}$ , independently of one another.

Determine the probability that $x^3 + y^3 + z^3$ is divisible by $9$ .

(4 marks)
Level 4 · 8 marksTry it
2
Prove, by exhaustion, that

$n^3 + 2$

is not a multiple of 9 for any integer $n$

(4 marks)
Level 4 · 4 marksTry it