Problem 28 · 2010 Math Kangaroo
Stretch
Number Theory
factorizationprimes
Along each side of a pentagon a positive integer is written. Numbers on adjacent sides never have a common factor bigger than 1, while numbers on non-adjacent sides always have a common factor bigger than 1. There are several possibilities for this situation, but one of the following numbers can never be on a side of the pentagon. Which one?
Show answer
Answer: C — 19
Show hints
Hint 1 of 2
Each side must share a factor with its two non-adjacent sides but none with its neighbours.
Still stuck? Show hint 2 →
Hint 2 of 2
A prime number on a side forces both its partner sides to be its multiples - and those partners are neighbours of each other.
Show solution
Approach: why a prime side is impossible
- A side's two non-adjacent partners must share its factor; but in a pentagon those two partners are next to each other.
- If the side were a prime, both partners would be multiples of that prime and so share it - yet as neighbours they must be coprime.
- The only prime offered is 19, so 19 can never be used.
Mark:
· log in to save