Difference between revisions of "2001 IMO Problems/Problem 6"
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First, <math>(KL+MN)-(KM+LN)=(K-N)(L-M)>0</math> as <math>K>N</math> and <math>L>M</math>. Thus, <math>KL+MN>KM+LN</math>. | First, <math>(KL+MN)-(KM+LN)=(K-N)(L-M)>0</math> as <math>K>N</math> and <math>L>M</math>. Thus, <math>KL+MN>KM+LN</math>. |
Revision as of 18:37, 17 March 2017
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 |