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WMA12/01 Jun 2021, Question 3: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 5 marks

The question, in short

Two proofs: for single digit primes p show p³ + p is divisible by 10, and using algebra show (n + 1)³ − n³ is not a multiple of 3 for n ∈ ℕ.

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Part (i) · 2 marks

Prove that p³ + p is a multiple of 10 for every single digit prime p.

  1. 1The single digit primes are 2, 3, 5 and 7
  2. 22³ + 2 = 10 and 3³ + 3 = 30
  3. 35³ + 5 = 130 and 7³ + 7 = 350
  4. 4All four are multiples of 10, so the result holds for every single digit prime

AnswerShown: p = 2: 10; p = 3: 30; p = 5: 130; p = 7: 350 — all multiples of 10, and 2, 3, 5, 7 are the only single-digit primes.

Part (ii) · 3 marks

Using algebra, show that (n + 1)³ − n³ is not a multiple of 3 for n ∈ ℕ.

  1. 1(n + 1)³ = n³ + 3n² + 3n + 1
  2. 2(n + 1)³ − n³ = 3n² + 3n + 1
  3. 3= 3(n² + n) + 1
  4. 43(n² + n) is a multiple of 3, so the expression leaves remainder 1 and is not a multiple of 3

AnswerShown: (n + 1)³ − n³ = 3n² + 3n + 1 = 3(n² + n) + 1, which is one more than a multiple of 3.

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