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Let a, b be nonzero integers and c be some integer. Which of the following statements must be true about the linear Diophantine equation below? ax + by = c (Recall that solution refers to integer solutions, i.e., pairs of integers x= xo, y = yo such that the equation above is satisfied.) If the greatest common divisor of a and b is also a divisor of c, then there must exist infinitely many solutions. All of the solutions of the Diophantine equation are of the form x = xo + bk, y=yo-ak, kez where x = xo, y = yo is a solution. □ Ifa, b have a common divisor that divides c, then there must be at least one solution. By Bezout's identity, the Diophantine equation only has a solution if c is equal to the greatest common divisor of a and b. If the Diophantine equation has a rational solution, i.e., a choice of rational numbers x= xo, y = yo that satisfies the equation, then the equation has at least one integer solution. If the greatest common divisor of a and b is also a divisor of c, then there exists at least one solution. □ If a, b have a common divisor that does not divide c, then there are no solutions.

Let a, b be nonzero integers and c be some integer. Which of the following statements must be true about the linear Diophantine equation below? ax + by = c (Recall that solution refers to integer solutions, i.e., pairs of integers x= xo, y = yo such that the equation above is satisfied.) If the greatest common divisor of a and b is also a divisor of c, then there must exist infinitely many solutions. All of the solutions of the Diophantine equation are of the form x = xo + bk, y=yo-ak, kez where x = xo, y = yo is a solution. □ Ifa, b have a common divisor that divides c, then there must be at least one solution. By Bezout's identity, the Diophantine equation only has a solution if c is equal to the greatest common divisor of a and b. If the Diophantine equation has a rational solution, i.e., a choice of rational numbers x= xo, y = yo that satisfies the equation, then the equation has at least one integer solution. If the greatest common divisor of a and b is also a divisor of c, then there exists at least one solution. □ If a, b have a common divisor that does not divide c, then there are no solutions.

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
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Need help with this Intro to Elementary Number Theory homework problem.

 

covered topics

  • prime numbers
  • linear Diophantine equations
  • systems of linear Diophantine equations.

 

Let a, b be nonzero integers and c be some integer. Which of the following statements must be true about the linear Diophantine equation below? ax + by = c (Recall that solution refers to integer solutions, i.e., pairs of integers x = xo, y = yo such that the equation above is satisfied.) If the greatest common divisor of a and b is also a divisor of c, then there must exist infinitely many solutions. All of the solutions of the Diophantine equation are of the form x = xo + bk, y yo - ak, kez 00 000 where x= xo, y = yo is a solution. If a, b have a common divisor that divides c, then there must be at least one solution. By Bezout's identity, the Diophantine equation only has a solution if c is equal to the greatest common divisor of a and b. If the Diophantine equation has a rational solution, i.e., a choice of rational numbers x = xo, y = yo that satisfies the equation, then the equation has at least one integer solution. If the greatest common divisor of a and b is also a divisor of c, then there exists at least one solution. If a, b have a common divisor that does not divide c, then there are no solutions.
Transcribed Image Text:Let a, b be nonzero integers and c be some integer. Which of the following statements must be true about the linear Diophantine equation below? ax + by = c (Recall that solution refers to integer solutions, i.e., pairs of integers x = xo, y = yo such that the equation above is satisfied.) If the greatest common divisor of a and b is also a divisor of c, then there must exist infinitely many solutions. All of the solutions of the Diophantine equation are of the form x = xo + bk, y yo - ak, kez 00 000 where x= xo, y = yo is a solution. If a, b have a common divisor that divides c, then there must be at least one solution. By Bezout's identity, the Diophantine equation only has a solution if c is equal to the greatest common divisor of a and b. If the Diophantine equation has a rational solution, i.e., a choice of rational numbers x = xo, y = yo that satisfies the equation, then the equation has at least one integer solution. If the greatest common divisor of a and b is also a divisor of c, then there exists at least one solution. If a, b have a common divisor that does not divide c, then there are no solutions.
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