Use the Euclidean Algorithm to express gcd(43201, 14234) as an integral linear combination of 43201, 14234

To express the greatest common divisor (gcd) of \( 43201 \) and \( 14234 \) as a linear combination of these two integers, we will use the **Euclidean Algorithm** and **Back Substitution**.

### Step 1: Apply the Euclidean Algorithm
The Euclidean Algorithm repeatedly divides the larger number by the smaller one and takes the remainder until the remainder is zero. The last non-zero remainder is the gcd.

1. \( 43201 \div 14234 = 3 \) with remainder \( 43201 - 3 \times 14234 = 510 \).

   Thus, \( 43201 = 3 \times 14234 + 510 \).

2. \( 14234 \div 510 = 27 \) with remainder \( 14234 - 27 \times 510 = 244 \).

   Thus, \( 14234 = 27 \times 510 + 244 \).

3. \( 510 \div 244 = 2 \) with remainder \( 510 - 2 \times 244 = 22 \).

   Thus, \( 510 = 2 \times 244 + 22 \).

4. \( 244 \div 22 = 11 \) with remainder \( 244 - 11 \times 22 = 2 \).

   Thus, \( 244 = 11 \times 22 + 2 \).

5. \( 22 \div 2 = 11 \) with remainder \( 22 - 11 \times 2 = 0 \).

   Thus, \( 22 = 11 \times 2 + 0 \).

Since the remainder is now 0, the **gcd is 2**.

### Step 2: Back Substitution
Now we will work backwards through the equations to express the gcd as a linear combination of 43201 and 14234.

1. From \( 244 = 11 \times 22 + 2 \), we get:
   \[
   2 = 244 - 11 \times 22
   \]
   
2. From \( 510 = 2 \times 244 + 22 \), we substitute \( 22 \):
   \[
   22 = 510 - 2 \times 244
   \]
   Substituting into the previous equation:
   \[
   2 = 244 - 11 \times (510 - 2 \times 244) = 244 - 11 \times 510 + 22 \times 244
   \]
   Simplifying:
   \[
   2 = 23 \times 244 - 11 \times 510
   \]

3. From \( 14234 = 27 \times 510 + 244 \), substitute \( 244 \):
   \[
   244 = 14234 - 27 \times 510
   \]
   Substitute into the previous equation:
   \[
   2 = 23 \times (14234 - 27 \times 510) - 11 \times 510 = 23 \times 14234 - 621 \times 510 - 11 \times 510
   \]
   Simplifying:
   \[
   2 = 23 \times 14234 - 632 \times 510
   \]

4. From \( 43201 = 3 \times 14234 + 510 \), substitute \( 510 \):
   \[
   510 = 43201 - 3 \times 14234
   \]
   Substitute into the previous equation:
   \[
   2 = 23 \times 14234 - 632 \times (43201 - 3 \times 14234) = 23 \times 14234 - 632 \times 43201 + 1896 \times 14234
   \]
   Simplifying:
   \[
   2 = 1919 \times 14234 - 632 \times 43201
   \]

### Final Expression
Thus, we have:
\[
\boxed{2 = 1919 \times 14234 - 632 \times 43201}
\]

This is the gcd expressed as a linear combination of 43201 and 14234.

Would you like further details or clarification?

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Here are 5 related questions to deepen your understanding:
1. How does the Euclidean Algorithm guarantee finding the gcd?
2. Why do we use back substitution in the Euclidean Algorithm?
3. Can the gcd of two numbers always be written as a linear combination?
4. How would the solution change if the gcd were 1?
5. What is the significance of the gcd in number theory?

**Tip**: The Euclidean Algorithm is efficient because it reduces the size of numbers quickly through division.

Learn how to use the Euclidean Algorithm to find the greatest common divisor (gcd) of 43201 and 14234, and express it as an integral linear combination. This detailed step-by-step solution includes back substitution and applies number theory concepts. Ideal for students in Grades 11-12.

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