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TMUA 2022 · Paper 2 · Question 3 of 20

TMUA 2022 Paper 2 Question 3

Proof and counterexample — Counterexamples. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Proof and counterexampleCounterexamples8 options
Consider the following statement about the positive integer n:

if n is prime, then n2 + 2 is not prime

Which of the following is a counterexample to this statement?

I   n = 2
II  n = 3
III n = 4

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
A counterexample needs the hypothesis to hold and the conclusion to fail. For n = 2, which is prime, n2 + 2 = 6 is not prime, so the statement holds. For n = 4 the hypothesis fails, since 4 is not prime. But n = 3 is prime and 32 + 2 = 11 is also prime, so the conclusion fails — only II.