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 = x0, 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 y yo ak, keZ x=x0+bk, 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 intege 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 = x0, 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 y yo ak, keZ x=x0+bk, 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 intege 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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Need help with ONLY the last four sub parts on this Intro to Elementary Number Theory homework problem.

covered topics

prime numbers
linear Diophantine equations
systems of linear Diophantine equations.

Let a, bbe 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 = x0, 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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