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TMUA 2016 · Paper 2 · Question 5 of 20

TMUA 2016 Paper 2 Question 5

Proof and counterexample — Counterexamples · numbers of the form 6k ± 1. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Proof and counterexampleCounterexamples · numbers of the form 6k ± 15 options
Consider the statement:

(*) A whole number n is prime if it is 1 less or 5 less than a multiple of 6.

How many counterexamples to (*) are there in the range 0 < n < 50?

  1. A2
  2. B3
  3. C4
  4. D5
  5. E6
Show the answer and worked solution
answer · C
  1. A2
  2. B3
  3. C4
  4. D5
  5. E6
A counterexample is a number of the stated form that is not prime. Numbers 1 less than a multiple of 6 are 5,  11,  17,  23,  29,  35,  41,  47; of these only 35 = 5×7 fails to be prime. Numbers 5 less than a multiple of 6 are 1,  7,  13,  19,  25,  31,  37,  43,  49; here 1 is not prime, and neither is 25 nor 49. That gives 35,  1,  25,  49 — four counterexamples. Forgetting that 1 is not a prime is the usual way to get 3.