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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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).]
%3D
1
%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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F871e9154-a0ff-45dd-bc27-ca399142d422%2Feb24162a-eb77-410c-84ee-e41446dcb942%2Foiqprtp_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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).]
%3D
1
%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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
