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TMUA 2020 · Paper 2 · Question 4 of 20

TMUA 2020 Paper 2 Question 4

Proof and counterexample — Counterexamples · sums of two non-primes. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Proof and counterexampleCounterexamples · sums of two non-primes8 options
Consider the following statement:

Every positive integer N that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1.

Which of the following is/are counterexample(s) to this statement?

I    N = 5
II   N = 7
III  N = 9

  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 · G
  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 has to satisfy the hypothesis and fail the conclusion, so N = 5 is disqualified at once: it is not greater than 6, and the statement says nothing about it. The non-prime integers greater than 1 start 4,  6,  8,  9,  10,  …. For N = 7 the only candidate pair is 4 + 3, and 3 is prime, so 7 fails the conclusion. For N = 9 the options are 4 + 5, 6 + 3 and 8 + 1, none of which works, so 9 fails too. Hence II and III.