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.

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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.