1984 IMO Problems/Problem 2
Contents
Problem
Find one pair of positive integers such that is not divisible by , but is divisible by .
Solution 1
So we want and , so we want . Now take e.g. and get . Now by some standard methods like Hensels Lemma (used to the polynomial , so seen as constant from now) we get also some with and , so and we are done. (in this case it gives )
This solution was posted and copyrighted by ZetaX. The original thread for this problem can be found here: [1]
Solution 2
Lemma: .
Proof: Recall that if , then .
Therefore, .
.
However, if , then .
So now, we need to find one pair of integers (a, b) such that . This means . . But this is true for all pairs of integers (a, b). So any random pair of integers would work.
Footnote: Even a pair of integers (a, b) which satisfies would work. So the condition given is irrelevant. Try it!
See Also
1984 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |