# 11 - (i The linear Diophantine Equation ax by = c has a...

This preview shows pages 1–3. Sign up to view the full content.

Theorem. Let a, b, c Z . If c | ab and gcd ( a, c ) = 1, then c | b . Proof. Since gcd ( a, c ) = 1, there exist x, y Z s.t. 1 = ax + cy. Then b = ( ab ) x + cby. Since c | ( ab ) and c | ( cb ), we have c | b. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Linear Diophantine Equations of one variable: ax = b, where a, b Z are given, a = 0, x Z is a variable. of two variables: ax + by = c, where a, b, c Z are given, a = 0 , b = 0, x, y Z are variables. 2
Linear Diophantine Equation Theorem.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (i) The linear Diophantine Equation ax + by = c has a solution if and only if gcd( a, b ) | c. (ii) If x = x and y = y is one particular solu-tion, then the complete solution is x = x + m b d , y = y-m a d for all m ∈ Z . 3...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern