Answered: Proof of CRT (for two moduli) We must…

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Proof of CRT (for two moduli) We must prove that f: Zm,ma Zmx Zm gimern by is a bijection.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.4: Mathematical Induction

Problem 40E

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Question

Prove the chinese remainder theorem by this steps in the picture , explain how the Pigeonhole principle gives us that the function is surjective once we know it is injective. The first part of the proof where the result is independent of the choice of representive why do that , there two rules * they show multiplication and Addition in Z_nare independent of choice of representative. In the proof where is that important.

* ((x MOD n) + (y MOD n)) MOD n = (x+y)MOD n

((x MOD n)(y MOD n)) MOD n = (xy) MOD n

Proof of CRT (for two modei)
We must prove that f: Zm,m Zmx Zmy givern by
is a
bijection.
Proof
O Fist check that f is well-delined, ie. that ļ is independent
of the choice of representadive x for the class [m
2 Show that f is one-to-one
Show that e is onto
This is trinial if f is one-tonone, as

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