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WMA12/01 Oct 2022, Question 1: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 3 marks

The question, in short

For positive integers a, b, c with c = b + 2 and a + b + c = 10, prove by exhaustion that abc is always even, using a table of a, b, c if helpful.

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Solution · 3 marks

  1. 1Substitute c = b + 2: a + b + (b + 2) = 10, so a = 8 − 2b
  2. 2a ≥ 1 needs 8 − 2b ≥ 1, so b can only be 1, 2 or 3
  3. 3b = 1: a = 6, c = 3, abc = 18
  4. 4b = 2: a = 4, c = 4, abc = 32
  5. 5b = 3: a = 2, c = 5, abc = 30
  6. 6All possible cases give an even product, so abc is always even

AnswerShown: the only cases are (a, b, c) = (6, 1, 3), (4, 2, 4), (2, 3, 5) with products 18, 32, 30 — all even, so abc is always even.

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Also searched as: WMA12 October 2022 · WMA12/01 · wma12 oct22 · IAS Maths Pure Mathematics 2 October 2022 · question 1.
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