Linear Diophantine Equations and Solutions

This video tutorial covers solving linear Diophantine equations, focusing on integer solutions. It begins with a historical overview of Diophantine equations, named after Diophantus of Alexandria. The tutorial then narrows down to linear Diophantine equations, explaining their form and how to solve them using an example equation. The process involves converting the equation to a congruence, simplifying it, and finding integer solutions. The tutorial concludes by demonstrating how to verify solutions with different values of k.

Who was the Greek mathematician known for studying equations with integer solutions?

• Archimedes

• Pythagoras

• Euclid

• Diophantus

What is a key characteristic of a Diophantine equation?

• It has no solutions.

• It has only rational solutions.

• It has only integer solutions.

• It has only real solutions.

What form does a linear Diophantine equation take?

• a^2 + b^2 = c^2

• ax^2 + bx + c = 0

• a/b = c/d

• ax + by = c

What is the first step in solving a linear Diophantine equation using congruences?

• Convert the equation to a congruence.

• Use the Pythagorean theorem.

• Apply the quadratic formula.

• Find the least common multiple.

What is the greatest common divisor of 51 and 87?

• 3

• 1

• 29

• 17

In the example, what is the simplified equation after dividing by the greatest common divisor?

• 51x + 87y = 123

• 17x + 29y = 41

• x + y = 1

• 29x + 17y = 41

What is the solution for y in the congruence 12y ≡ 7 (mod 17)?

• y = 2 + 17k

• y = 4 + 17k

• y = 1 + 17k

• y = 3 + 17k

What is the expression for x in terms of k after solving the example equation?

• x = 1 + 29k

• x = -1 - 29k

• x = 2 + 29k

• x = -2 - 29k

If k = 0, what are the values of x and y in the example solution?

• x = -1, y = 2

• x = 1, y = -2

• x = 2, y = -1

• x = 0, y = 0

For k = 3, what are the values of x and y in the example solution?

• x = -88, y = 53

• x = -57, y = 32

• x = 88, y = -53

• x = 57, y = -32