Question number 27. The sides of a pentagon are labeled with natural numbers in such a way that the greatest common divisor of any two adjacent sides is 1 and the greatest common divisor of any two non-adjacent sides is greater than 1. Which of these could not have been used as a label? So in my example pentagon, I will label three of the sides here as A, B, and C, and then looking at the adjacent sides here, for example, A is adjacent to B, what we see is A and B are what we call relatively prime, which means exactly what the problem says, that only the number 1 divides both at the same time. So one way to ensure that is to set A to be a prime number, and we have one prime number here in our possibilities, and that's 11. So if A is prime, then we know that A will be relatively prime to its neighbors. or I should say to any smaller number, to be exact. For example, b equal to 22 is not relatively prime to a equal to 11. But as long as we choose b and c to be smaller than 11, then those numbers will be relatively prime to 11. And so that pretty much answers our question. Because if we then look at non-adjacent sides here, and if a is equal to 11, but c is equal to, for example, 1 or 2 or 3, anything up to 10, then a and c are relatively prime. Only the number 1 will divide both of them at the same time. And so according to our problem, this is not allowed, the greatest common divisor of any two non-adjacent sides has to be greater than 1. So if a is 11 and c is any number less than 11, this is not the case. So that causes trouble in our problem. And so because of this divisibility condition, prime numbers have to be avoided. So we choose e here for the answer. This could not have been used as a label.

pentagon labeling

greatest common divisor

natural numbers

prime numbers

non-adjacent sides