Difference between revisions of "2017 USAJMO Problems/Problem 1"
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where we use our lemma and the Euler totient theorem: <math>a^{\phi{n}} \equiv 1 \pmod{n}</math> when <math>a</math> and <math>n</math> are relatively prime. | where we use our lemma and the Euler totient theorem: <math>a^{\phi{n}} \equiv 1 \pmod{n}</math> when <math>a</math> and <math>n</math> are relatively prime. | ||
+ | ==Solution 2== | ||
+ | Let <math>x</math> be any odd number above 1 | ||
{{MAA Notice}} | {{MAA Notice}} | ||
==See also== | ==See also== | ||
{{USAJMO newbox|year=2017|beforetext=|before=First Problem|num-a=2}} | {{USAJMO newbox|year=2017|beforetext=|before=First Problem|num-a=2}} |
Revision as of 19:27, 19 April 2017
Contents
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 |