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
The positive integers $a$ and $b$ satisfy

$a < b \qquad \text{and} \qquad ab = 30$

Prove, by exhaustion, that $a + b$ is always a prime number.

The table below may be used to set out your cases. You do not have to fill every row.
$a$ $b$ $a + b$
(3 marks)
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$a$	$b$	$a + b$