2. Let n₁,...,nk be pairwise relatively prime positive integers, and let n = N₁ ··· Nk.Recall we proved in class that for any integers a₁,..., ak, there exists a uniquex € Zn such thatx = a (mod ni)ifor all i = {1 k}. In this problem we will learn how to find the solution .9 ... 9(a) For each i = {1, ..., k}, since gcd (ni, n/ni) 1, one can find (using the Eu-clidean algorithm) integers b; and c; such that bini + ci(n/ni) = 1. Letci(n/ni).Show thatmį = 1 - binį=x = a₁x = a₂x = az=X x = a₁m₁ ++ акткis the unique solution in Zn such that x = a; (mod nį) for all i = {1,..., k}.(b) Solve the system of congruence equations(mod 3),(mod 5),(mod 7).

this is also famous known as the chinese remainder theorem

2. Let n₁,..., nk be pairwise relatively prime positive integers, and let n = n₁ ... nk. Recall we proved in class that for any integers a₁,..., ak, there exists a unique EZ such that x = ai (mod n₁) for all i E {1, ..., k}. In this problem we will learn how to find the solution à. (a) For each i = {1,..., k}, since gcd(ni, n/ni) 1, one can find (using the Eu- clidean algorithm) integers b; and c; such that bini +ci(n/ni) = 1. Let Show that - Mi = 1 bini= ci(n/ni). = x = a₁m₁ + + ak mk is the unique solution in Zn such that x =ai (mod nį) for all i = {1, ..., k}. (b) Solve the system of congruence equations x = a1 x = a₂ x = a3 ... (mod 3), (mod 5), (mod 7).

2. Let n₁,..., nk be pairwise relatively prime positive integers, and let n = n₁ ... nk. Recall we proved in class that for any integers a₁,..., ak, there exists a unique EZ such that x = ai (mod n₁) for all i E {1, ..., k}. In this problem we will learn how to find the solution à. (a) For each i = {1,..., k}, since gcd(ni, n/ni) 1, one can find (using the Eu- clidean algorithm) integers b; and c; such that bini +ci(n/ni) = 1. Let Show that - Mi = 1 bini= ci(n/ni). = x = a₁m₁ + + ak mk is the unique solution in Zn such that x =ai (mod nį) for all i = {1, ..., k}. (b) Solve the system of congruence equations x = a1 x = a₂ x = a3 ... (mod 3), (mod 5), (mod 7).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 33EQ

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