IMC 2006 Q3
3. Which of the following is not prime?
- A \(2^2 - 1\)
- B \(2^3 - 1\)
- C \(2^5 - 1\)
- D \(2^6 - 1\)
- E \(2^7 - 1\)
Intermediate Mathematical Challenge 2006, Question 3. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(2^6 - 1\)
The values of the given options are 3, 7, 31, 63 and 127 respectively. All are prime, except 63. (Note that \(2^c - 1\) is never prime if \(c\) is composite. However, if \(p\) is prime, \(2^p - 1\) is not necessarily prime. The smallest example of this occurs for \(p = 11\): \(2^{11} - 1 = 23 \times 89\).)
Solution © UK Mathematics Trust, from the IMC 2006 solutions.