2017 USAJMO Problems/Problem 1
Prove that there are infinitely many distinct pairs of relatively prime 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.
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