- Analyze the linear Diophantine equation \(7x - 19y = 1\), where \(a=7, b=-19\).
- Use the Extended Euclidean Algorithm to find a particular solution by working backwards from the GCD process: \(1 = 3 \times 19 - 8 \times 7\).
- Identify the particular solution \(x_0 = -8\) and \(y_0 = -3\).
- State the general solution using an integer parameter \(k\): \(x = -8 - 19k, y = -3 - 7k\).
- \(\boxed{x = -8 - 19k, y = -3 - 7k}\) for \(k \in \mathbb{Z}\).
Explanation
- Problem Analysis and Approach
The problem asks to find the general solution to the linear Diophantine equation \(7x - 19y = 1\). This is an equation of the form \(ax + by = c\), where \(a = 7\), \(b = -19\), and \(c = 1\). Since the greatest common divisor \(\gcd(7, 19) = 1\), and \(1\) divides \(1\), integer solutions exist. Our goal is to find a particular solution \((x_0, y_0)\) using the Extended Euclidean Algorithm, and then express the general solution using an integer parameter \(k\).
- Extended Euclidean Algorithm for Particular Solution
First, we apply the Euclidean algorithm to the coefficients \(7\) and \(19\):
- \(19 = 2 \times 7 + 5\)
- \(7 = 1 \times 5 + 2\)
- \(5 = 2 \times 2 + 1\)
Now, we work backwards to express \(1\) as a linear combination of \(7\) and \(19\):
\(1 = 5 - 2 \times 2\)
Substituting \(2 = 7 - 1 \times 5\):
\(1 = 5 - 2 \times (7 - 5) = 3 \times 5 - 2 \times 7\)
Substituting \(5 = 19 - 2 \times 7\):
\(1 = 3 \times (19 - 2 \times 7) - 2 \times 7 = 3 \times 19 - 8 \times 7\)
Rearranging to match the form \(7x - 19y = 1\):
\(7(-8) + 19(3) = 1 \Rightarrow 7(-8) - 19(-3) = 1\).
A particular solution is \(x_0 = -8\) and \(y_0 = -3\).
- Deriving the General Solution
The general solution for a linear Diophantine equation \(ax + by = c\) is given by \(x = x_0 + \frac{b}{\gcd(a,b)} k\) and \(y = y_0 - \frac{a}{\gcd(a,b)} k\), where \(k \in \mathbb{Z}\).
Substituting our values \(a = 7\), \(b = -19\), \(c = 1\), \(x_0 = -8\), \(y_0 = -3\), and \(\gcd(7, 19) = 1\):
\(x = -8 + \frac{-19}{1} k = -8 - 19k\)
\(y = -3 - \frac{7}{1} k = -3 - 7k\)
Thus, the general integer solutions are \(x = -8 - 19k\) and \(y = -3 - 7k\) for any integer \(k\).
- Final Answer
The general solution to the equation \(7x - 19y = 1\) is given by the set of integer pairs \((x, y)\) such that \(x = -8 - 19k\) and \(y = -3 - 7k\), where \(k\) is any integer. We can verify this: \(7(-8 - 19k) - 19(-3 - 7k) = -56 - 133k + 57 + 133k = 1\). The solution is consistent. Therefore, the general solution is \(\boxed{x = -8 - 19k, y = -3 - 7k}\) for \(k \in \mathbb{Z}\).
Examples
Linear Diophantine equations are frequently used in cryptography and computer science. For example, in RSA encryption, one needs to find a modular multiplicative inverse, which is essentially solving a linear Diophantine equation \(ax + my = 1\). If you have two different cycles of tasks, like one repeating every 7 days and another every 19 days, finding when they align or calculating specific synchronization points involves solving equations of this type. Mastery of this method allows for precise scheduling in complex systems.