Confirm the following properties of the greatest common divisor:(a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1.[Hint: Because 1 = ax + by = au + cv for some x, y, u, v,(ax + by)(au + cv) = a(aux + cvx + byu) + bc(yv).]%3D1%3D%3D(b) If gcd(a , b) = 1, and c | a, then gcd(b , c) = 1.(c) If gcd(a , b) = 1, then gcd(ac , b) = gcd(c , b).(d) If gcd(a , b) = 1, and c | a + b, then gcd(a , c) = gcd(b , c) = 1.[Hint: Let d = gcd(a , c). Then d | a, d | c implies that d | (a + b) – a(e) If gcd(a , b) = 1, d | ac, and d |bc, then d | c.(f) If gcd(a , b) = 1, then gcd(a² , b²) = 1.[Hint: First show that ged(a , b²) = ged(a² , b) = 1.]%3D%3D%3D%3D%3D%3D%3D

according to our guidelines we can answer only three subparts, or first question and rest can be…

Confirm the following properties of the greatest common divisor: (a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1. [Hint: Because 1 = ax + by = au + cv for some x, y, u, v, 1 = (ax + by)(au + cv) = a(aux + cvx + byu) + bc(yv).] (b) If gcd(a , b) = 1, and c |a, then gcd(b , c) = 1. (c) If gcd(a , b) = 1, then gcd(ac , b) = gcd(c , b). %3D %3D %3D %3D

Confirm the following properties of the greatest common divisor: (a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1. [Hint: Because 1 = ax + by = au + cv for some x, y, u, v, 1 = (ax + by)(au + cv) = a(aux + cvx + byu) + bc(yv).] (b) If gcd(a , b) = 1, and c |a, then gcd(b , c) = 1. (c) If gcd(a , b) = 1, then gcd(ac , b) = gcd(c , b). %3D %3D %3D %3D

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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