Linear Diophantine Equation/Examples/17x + 19y = 23
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Example of Linear Diophantine Equation
The linear diophantine equation:
- $17 x + 19 y = 23$
has the general solution:
- $\tuple {x, y} = \tuple {207 + 19 t, -184 - 17 t}$
Graphical Presentation
Proof
Using the Euclidean Algorithm:
\(\text {(1)}: \quad\) | \(\ds 19\) | \(=\) | \(\ds 1 \times 17 + 2\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 17\) | \(=\) | \(\ds 8 \times 2 + 1\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds 2\) | \(=\) | \(\ds 2 \times 1\) |
Thus we have that:
- $\gcd \set {17, 19} = 1$
which is (trivially) a divisor of $23$.
So, from Solution of Linear Diophantine Equation, a solution exists.
Next we find a single solution to $17 x + 19 y = 23$.
Again with the Euclidean Algorithm:
\(\ds 1\) | \(=\) | \(\ds 17 - 8 \times 2\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 17 - 8 \times \paren {19 - 1 \times 17}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 9 \times 17 - 8 \times 19\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 23\) | \(=\) | \(\ds 23 \times \paren {9 \times 17 - 8 \times 19}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 207 \times 17 - 184 \times 19\) |
and so:
\(\ds x_0\) | \(=\) | \(\ds 207\) | ||||||||||||
\(\ds y_0\) | \(=\) | \(\ds -184\) |
is a solution.
From Solution of Linear Diophantine Equation, the general solution is:
- $\forall t \in \Z: x = x_0 + \dfrac b d t, y = y_0 - \dfrac a d t$
where $d = \gcd \set {a, b}$.
In this case:
\(\ds x_0\) | \(=\) | \(\ds 207\) | ||||||||||||
\(\ds y_0\) | \(=\) | \(\ds -184\) | ||||||||||||
\(\ds a\) | \(=\) | \(\ds 17\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds 19\) | ||||||||||||
\(\ds d\) | \(=\) | \(\ds 1\) |
giving:
\(\ds x\) | \(=\) | \(\ds 207 + 19 t\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds -184 - 17 t\) |
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-3}$ The Linear Diophantine Equation: Exercise $1 \ \text {(b)}$