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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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 x € Zn such that x = a (mod ni) i for 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. Let ci(n/ni). Show that mį = 1 - binį = x = a₁ x = a₂ x = az = X x = a₁m₁ + + актк is 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).
Transcribed Image Text: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 x € Zn such that x = a (mod ni) i for 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. Let ci(n/ni). Show that mį = 1 - binį = x = a₁ x = a₂ x = az = X x = a₁m₁ + + актк is 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).
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