2017 USAJMO Problems/Problem 2
Prove that there are infinitely many distinct pairs of relatively prime positive integers and such that is divisible by .
Let and . We see that and are relatively prime (they are consecutive positive odd integers).
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.
|2017 USAJMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5 • 6|
|All USAJMO Problems and Solutions|