Puzzle on the course “Theory of the numbers”

Puzzle on the course “Theory of the numbers”

Theorem

If integers a, b, a+b and r are mutually-prime, then there is such d, relatively prime with r, that the ends of the numbers ad and bd are equal on the module r.

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[Consequence. With relatively prime a, b, a+b and r, where the value r is undertaken from the equality:

1*) [math]a^n+b^n=(a+b)r^n=c^n[/math] or [math]a^n+b^n=(a+b)nr^n=c^n[/math], equality 1* is contradictory in the base r, since in the equality

2*) [math](ad)^n+(bd)^n=(cd)^n[/math] ([math]=Pr[/math]) right side is divided by r, but leftist is not divided.]

Proof is located in the stage of formulation.