WMA12/01 Jun 2025, Question 10: Proof by Exhaustion and Counter-example
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 4 marks
The question, in short
Using algebra and proof by exhaustion, show that m² + 3m + 2 is a multiple of 3 for every positive integer m not divisible by 3.
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Solution · 4 marks
- 1A positive integer that is not a multiple of 3 is m = 3p + 1 or m = 3p + 2 (p ≥ 0 an integer)
- 2m = 3p + 1: (3p + 1)² + 3(3p + 1) + 2 = 9p² + 6p + 1 + 9p + 3 + 2 = 9p² + 15p + 6 = 3(3p² + 5p + 2)
- 3m = 3p + 2: (3p + 2)² + 3(3p + 2) + 2 = 9p² + 12p + 4 + 9p + 6 + 2 = 9p² + 21p + 12 = 3(3p² + 7p + 4)
- 4Both cases are multiples of 3, so m² + 3m + 2 is always a multiple of 3
AnswerShown: m = 3p + 1 gives 3(3p² + 5p + 2) and m = 3p + 2 gives 3(3p² + 7p + 4), so m² + 3m + 2 is always a multiple of 3
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