Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Proposition 5.5.20. Given a modular equation ax = c (mod b), where a,b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b.

Proposition 5.5.20. Given a modular equation ax = c (mod b), where a,b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
See similar textbooks
icon
Related questions
Question

Please do Question 5.5.21 and the hint is to use Proposition 5.5.16. Please show everything and explain each 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 a', b'. By basic algebra, we have an + bm = this back in the original Diophantine equation, we get: da' and b = db' for some integers d(a'n + b'm). If we substitute d(a'n + b'm) = It follows that c is a multiple of, d, which is the gcd of a and b.
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 a', b'. By basic algebra, we have an + bm = this back in the original Diophantine equation, we get: da' and b = db' for some integers d(a'n + b'm). If we substitute d(a'n + b'm) = It follows that c is a multiple of, d, which is the gcd of a and b.
Proposition 5.5.20. Given a modular equation ax = c (mod b), where a, b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b. Exercise 5.5.21. Prove both the "if" and the "only if" parts of Proposi- tion 5.5.20. (*Hint*)
Transcribed Image Text:Proposition 5.5.20. Given a modular equation ax = c (mod b), where a, b, c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b. Exercise 5.5.21. Prove both the "if" and the "only if" parts of Proposi- tion 5.5.20. (*Hint*)
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra
College Algebra
Algebra
ISBN:
9781337282291
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS