Final Exam, Spring 2007, Question 2a

Question:

Prove that there cannot be three consecutive odd integers greater than or equal to 5 that are all prime.

Answer:

Given an odd integer k >= 5, need to show that at least one of k, k+2, k+4 is not prime.
It suffices to show that one of these numbers is divisible by 3.

Assume k is not a multiple of three. Then either k-1 or k+1 is a multiple of three.
If k-1 is divisible by three then k+2 is also divisible by three, so it is not prime.
If k+1 is divisible by three then k+4 is also divisible by three, so it is not prime.