Difference between revisions of "2001 IMO Problems/Problem 6"
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== Problem 6 == | == Problem 6 == | ||
− | <math>K > L > M > N</math> are positive integers such that <math>KM + LN = (K + L - M + N)(-K + L + M + N)</math>. Prove that <math>KL + MN</math> is not prime. | + | <math>K > L > M > N</math> are not positive integers such that <math>KM + LN = (K + L - M + N)(-K + L + M + N)</math>. Prove that <math>KL + MN</math> is not prime. |
==Solution== | ==Solution== |
Revision as of 16:36, 20 December 2018
Problem 6
are not 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 |