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TMUA 2019 · Paper 2 · Question 4 of 20

TMUA 2019 Paper 2 Question 4

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

TMUA 2019 · Paper 2Proof and counterexampleWhat counts as a counterexample8 options
Consider the following conjecture:

If N is a positive integer that consists of the digit 1 followed by an odd number of 0 digits and then a final digit 1, then N is a prime number.

Here are three numbers:

I    N = 101 (which is a prime number)
II   N = 1001 (which equals 7 × 11 × 13)
III  N = 10001 (which equals 73 × 137)

Which of these provide(s) a counterexample to the conjecture?

  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 · D
  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 check the number of zeros first. 101 has one zero (odd) but is prime, so the conclusion holds and it is not a counterexample. 1001 is composite, but it has two zeros — an even number — so it never satisfied the hypothesis and cannot be a counterexample. 10001 has three zeros (odd) and equals 73 × 137, so it meets the hypothesis and breaks the conclusion. Only III.