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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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 33EQ
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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.
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