WMA12/01 January 2024 Question 8: Proof by Exhaustion and Counter-example

Question 1

Find a natural number n for which n² + 3n + 1 is not prime, showing the statement is false.

Accepted Answer

Try n = 6: 6² + 3 × 6 + 1 = 36 + 18 + 1 = 55 55 = 5 × 11, so 55 is not prime So the statement is false Answer: Shown: n = 6 gives 6² + 3(6) + 1 = 55 = 5 × 11, which is not prime, so the statement is false.

Question 2

Using algebra, prove by exhaustion over all cases of n that n² − 2 is not a multiple of 4.

Accepted Answer

Every natural number is even (n = 2p) or odd (n = 2p + 1) Even: (2p)² − 2 = 4p² − 2, which leaves remainder 2 when divided by 4 Odd: (2p + 1)² − 2 = 4p² + 4p + 1 − 2 = 4(p² + p) − 1 4(p² + p) − 1 is one less than a multiple of 4 So in both cases n² − 2 is not a multiple of 4 Answer: Shown: n even, n = 2p: n² − 2 = 4p² − 2 = 4(p²) − 2, not a multiple of 4; n odd, n = 2p + 1: n² − 2 = 4p² + 4p − 1 = 4(p² + p) − 1, one less than a multiple of 4. Every n is even or odd, so n² − 2 is never a multiple of 4.

WMA12/01 Jan 2024, Question 8: Proof by Exhaustion and Counter-example

The question, in short

Part (i) · 2 marks

Part (ii) · 4 marks

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