Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Exercise 5.5.17. Prove the "if" part of Proposition 5.5.16. *Hint

Exercise 5.5.17. Prove the "if" part of Proposition 5.5.16. *Hint

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
See similar textbooks
icon
Related questions
Question

Please do exercise 5.5.17 and the hints states to use Proposition 5.5.8 and please do step by step

Proposition 5.5.16. Given the Diophantine equation an + bm = c, where a, b, c are integers. Then the equation has integer solutions for n and m if and only if c is a multiple of the gcd of a and b. PROOF. Since this is an "if and only if" proof, we need to prove it both ways. We'll do "only if" here, and leave the other way as an exercise. Since we're doing the "only if" part, we assume that an + bm = c is solvable. We'll represent the gcd of a and b by the letter d. Since gcd(a, b) divides both a and b, we may write a = da' and b = db' for some integers a', b'. By basic algebra, we have an + bm = d(a'n + b'm). If we substitute this back in the original Diophantine equation, we get: d(a'n + b'm) = c It follows that c is a multiple of, d, which is the gcd of a and b. Exercise 5.5.17. Prove the "if" part of Proposition 5.5.16. (*Hint*)
Transcribed Image Text:Proposition 5.5.16. Given the Diophantine equation an + bm = c, where a, b, c are integers. Then the equation has integer solutions for n and m if and only if c is a multiple of the gcd of a and b. PROOF. Since this is an "if and only if" proof, we need to prove it both ways. We'll do "only if" here, and leave the other way as an exercise. Since we're doing the "only if" part, we assume that an + bm = c is solvable. We'll represent the gcd of a and b by the letter d. Since gcd(a, b) divides both a and b, we may write a = da' and b = db' for some integers a', b'. By basic algebra, we have an + bm = d(a'n + b'm). If we substitute this back in the original Diophantine equation, we get: d(a'n + b'm) = c It follows that c is a multiple of, d, which is the gcd of a and b. Exercise 5.5.17. Prove the "if" part of Proposition 5.5.16. (*Hint*)
Proposition 5.5.8. The gcd of two numbers a and b can be written in the form n · a + m ·b where n and m are integers.
Transcribed Image Text:Proposition 5.5.8. The gcd of two numbers a and b can be written in the form n · a + m ·b where n and m are integers.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS