To find the greatest common divisor (gcd) of $143$ and $227$ using the Euclidean Algorithm, follow these steps:

1. Divide 227 by 143 and write the quotient and remainder:

$227 \div 143 = 1 \quad \text{(quotient)}$ $227 - 1 \times 143 = 227 - 143 = 84 \quad \text{(remainder)}$

So, $227 = 143 \times 1 + 84$.

2. Divide 143 by 84 and find the quotient and remainder:

$143 \div 84 = 1 \quad \text{(quotient)}$ $143 - 1 \times 84 = 143 - 84 = 59 \quad \text{(remainder)}$

So, $143 = 84 \times 1 + 59$.

3. Divide 84 by 59 and find the quotient and remainder:

$84 \div 59 = 1 \quad \text{(quotient)}$ $84 - 1 \times 59 = 84 - 59 = 25 \quad \text{(remainder)}$

So, $84 = 59 \times 1 + 25$.

4. Divide 59 by 25 and find the quotient and remainder:

$59 \div 25 = 2 \quad \text{(quotient)}$ $59 - 2 \times 25 = 59 - 50 = 9 \quad \text{(remainder)}$

So, $59 = 25 \times 2 + 9$.

5. Divide 25 by 9 and find the quotient and remainder:

$25 \div 9 = 2 \quad \text{(quotient)}$ $25 - 2 \times 9 = 25 - 18 = 7 \quad \text{(remainder)}$

So, $25 = 9 \times 2 + 7$.

6. Divide 9 by 7 and find the quotient and remainder:

$9 \div 7 = 1 \quad \text{(quotient)}$ $9 - 1 \times 7 = 9 - 7 = 2 \quad \text{(remainder)}$

So, $9 = 7 \times 1 + 2$.

7. Divide 7 by 2 and find the quotient and remainder:

$7 \div 2 = 3 \quad \text{(quotient)}$ $7 - 3 \times 2 = 7 - 6 = 1 \quad \text{(remainder)}$

So, $7 = 2 \times 3 + 1$.

8. Divide 2 by 1 and find the quotient and remainder:

$2 \div 1 = 2 \quad \text{(quotient)}$ $2 - 2 \times 1 = 2 - 2 = 0 \quad \text{(remainder)}$

So, $2 = 1 \times 2 + 0$.

At this point, the remainder is $0$, and the divisor at this step is $1$, which means the gcd is 1.

Thus, $\gcd(143, 227) = 1$.

Would you like a more detailed breakdown or explanation of any step? Feel free to ask!

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Here are some related questions you might find interesting:

1. What does it mean when the gcd of two numbers is 1?
2. How can the Euclidean algorithm be used to solve Diophantine equations?
3. What is the significance of the gcd in number theory?
4. Can you explain how the Euclidean algorithm applies to polynomials?
5. What is the extended Euclidean algorithm, and how is it different from the regular Euclidean algorithm?

Tip: The Euclidean Algorithm is a very efficient way to find the gcd of two numbers, especially when the numbers are large.