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WMA12/01 Oct 2023, 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 = 3a + 1 and a + b + c = 15, use exhaustion to prove abc is always a multiple of 4, optionally with a table of a, b, c, abc.

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Solution · 3 marks

  1. 1Substitute c = 3a + 1 into a + b + c = 15: 4a + b = 14, so b = 14 − 4a
  2. 2b > 0 needs a < 3.5, so a = 1, 2 or 3
  3. 3a = 1: b = 10, c = 4, abc = 40 = 4 × 10
  4. 4a = 2: b = 6, c = 7, abc = 84 = 4 × 21
  5. 5a = 3: b = 2, c = 10, abc = 60 = 4 × 15
  6. 6Every possible case gives a multiple of 4, so abc is always a multiple of 4

AnswerShown: the only cases are (a, b, c) = (1, 10, 4), (2, 6, 7), (3, 2, 10), giving abc = 40, 84, 60 = 4×10, 4×21, 4×15, so abc is always a multiple of 4

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Also searched as: WMA12 October 2023 · WMA12/01 · wma12 oct23 · IAS Maths Pure Mathematics 2 October 2023 · question 1.
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