Question

13x + 11y = 1

Answer 1

The equation \( 13x + 11y = 1 \) is a linear Diophantine equation, which means we are looking for integer solutions for \( x \) and \( y \).

To find a particular solution, we can use the Extended Euclidean Algorithm to express \( 1 \) as a linear combination of \( 13 \) and \( 11 \).

### Step 1: Apply Euclidean Algorithm
First, we can find the greatest common divisor (gcd) of \( 13 \) and \( 11 \):

\[
13 = 1 \cdot 11 + 2
\]
\[
11 = 5 \cdot 2 + 1
\]
\[
2 = 2 \cdot 1 + 0
\]

Thus, \( \gcd(13, 11) = 1 \).

### Step 2: Back Substitute to find coefficients
Now, we backtrack to express \( 1 \) as a combination of \( 13 \) and \( 11 \):

1. From the second step above:
\[
1 = 11 - 5 \cdot 2
\]
2. Substitute \( 2 \) from the first step:
\[
2 = 13 - 1 \cdot 11
\]
Putting this into the previous equation gives:
\[
1 = 11 - 5(13 - 1 \cdot 11) = 11 - 5 \cdot 13 + 5 \cdot 11 = 6 \cdot 11 - 5 \cdot 13
\]

Thus, one particular solution to \( 13x + 11y = 1 \) is \( x = -5 \) and \( y = 6 \).

### Step 3: General solution
The general solution to the equation \( 13x + 11y = 1 \) can be expressed as:
\[
x = -5 + 11n
\]
\[
y = 6 - 13n
\]
where \( n \) is any integer.

### Example Solutions
- For \( n = 0 \): \( (x, y) = (-5, 6) \)
- For \( n = 1 \): \( (x, y) = (6, -7) \)
- For \( n = -1 \): \( (x, y) = (-16, 19) \)

You can choose any integer \( n \) to generate different integer solutions to the equation.