Using the Euclidean Algorithm (E.A.) and back substitution to find integer/general solutions for 
the following if it exists: 
26046 * 𝑥 + 12762 * 𝑦 = 18 

To solve the Diophantine equation \( 26046x + 12762y = 18 \) using the Euclidean Algorithm (E.A.) and back substitution, we will proceed step by step.

### Step 1: Apply the Euclidean Algorithm

We first use the Euclidean Algorithm to find the greatest common divisor (gcd) of 26046 and 12762. If the gcd divides 18, then a solution exists.

Perform the Euclidean Algorithm by repeatedly dividing the larger number by the smaller number and taking remainders:

1. \( 26046 \div 12762 = 2 \) with remainder \( 26046 - 2 \times 12762 = 530 \)
2. \( 12762 \div 530 = 24 \) with remainder \( 12762 - 24 \times 530 = 2 \)
3. \( 530 \div 2 = 265 \) with remainder \( 530 - 265 \times 2 = 0 \)

The gcd is 2, because the last non-zero remainder is 2.

Since 2 divides 18 (i.e., \( \gcd(26046, 12762) = 2 \)), a solution exists.

### Step 2: Solve the equation for gcd = 2 using back substitution

We now express the gcd (2) as a linear combination of 26046 and 12762 by back substitution. From the Euclidean algorithm steps:

1. \( 2 = 12762 - 24 \times 530 \)
2. Substitute \( 530 = 26046 - 2 \times 12762 \) into the equation:
   \[
   2 = 12762 - 24 \times (26046 - 2 \times 12762)
   \]
   Simplifying:
   \[
   2 = 12762 - 24 \times 26046 + 48 \times 12762 = 49 \times 12762 - 24 \times 26046
   \]
   Thus:
   \[
   2 = -24 \times 26046 + 49 \times 12762
   \]

### Step 3: Multiply the equation by 9

We now need to find a solution for the original equation \( 26046x + 12762y = 18 \). Since \( 2 = -24 \times 26046 + 49 \times 12762 \), multiply both sides of this equation by 9 to get:

\[
18 = 9 \times (-24 \times 26046 + 49 \times 12762)
\]
\[
18 = (-216) \times 26046 + 441 \times 12762
\]

Thus, one particular solution is \( x = -216 \) and \( y = 441 \).

### Step 4: General solution

The general solution to the Diophantine equation \( 26046x + 12762y = 18 \) is given by:

\[
x = -216 + \frac{12762}{\gcd(26046, 12762)} \cdot t = -216 + 6381t
\]
\[
y = 441 - \frac{26046}{\gcd(26046, 12762)} \cdot t = 441 - 13023t
\]

where \( t \) is any integer.

### Final Answer:

The integer solutions are:

\[
x = -216 + 6381t
\]
\[
y = 441 - 13023t
\]

where \( t \) is an integer.

---

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the general form of a Diophantine equation?
2. How does the Euclidean Algorithm help in solving Diophantine equations?
3. Why must the gcd divide the constant term for a solution to exist?
4. How do you find the general solution to a linear Diophantine equation?
5. Can a Diophantine equation have no solutions?

**Tip**: When solving Diophantine equations, always check if the gcd of the coefficients divides the constant term to ensure a solution exists.

Using the Euclidean Algorithm (E.A.) and back substitution to find integer/general solutions for the following if it exists: 26046 * 𝑥 + 12762 * 𝑦 = 18

Explore the step-by-step solution of the Diophantine equation 26046x + 12762y = 18 using the Euclidean Algorithm and back substitution. Learn key mathematical concepts, including how to express the greatest common divisor as a linear combination and derive the general integer solutions. Suitable for college-level mathematics or advanced high school students.

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