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==See also== | ==See also== | ||
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Revision as of 18:27, 19 April 2017
Contents
[hide]Problem
Prove that there are infinitely many distinct pairs of relatively prime integers
and
such that
is divisible by
.
Solution 1
Let and
. We see that
and
are relatively prime (they are consecutive positive odd integers).
Lemma: .
Since every number has a unique modular inverse, the lemma is equivalent to proving that . Expanding, we have the result.
Substituting for and
, we have
where we use our lemma and the Euler totient theorem:
when
and
are relatively prime.
Solution 2
Let be any odd number above 1
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2017 USAJMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |