2001 IMO Problems/Problem 6
Problem 6
are positive integers such that . Prove that is not prime.
Solution
First, as and . Thus, .
Similarly, since and . Thus, .
Putting the two together, we have
Now, we have: So, we have: Thus, it follows that Now, since if is prime, then there are no common factors between the two. So, in order to have we would have to have This is impossible as . Thus, must be composite.
See also
2001 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |