# Puzzle on the course “Theory of the numbers”

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

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

Proof is located in the stage of formulation.

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