a = 3, b=13. find the integers k and l such that gcd(a,b) = ak+bl

To find integers \( k \) and \( l \) such that \( \gcd(a, b) = ak + bl \), we can use the **Extended Euclidean Algorithm**. 

Here, we are given:
\[
a = 3, \quad b = 13
\]

### Step 1: Apply the Euclidean Algorithm
The Euclidean algorithm helps us find the greatest common divisor (gcd) of \( a \) and \( b \). The steps are as follows:

\[
13 = 3 \times 4 + 1 \quad \text{(divide 13 by 3, remainder is 1)}
\]
\[
3 = 1 \times 3 + 0 \quad \text{(divide 3 by 1, remainder is 0)}
\]
Since the remainder is 0, we conclude that:
\[
\gcd(3, 13) = 1
\]

### Step 2: Apply the Extended Euclidean Algorithm
Now we work backwards to express 1 as a linear combination of \( a \) and \( b \).

From the first equation:
\[
1 = 13 - 3 \times 4
\]
So we have:
\[
1 = 13 - 4 \times 3
\]

Thus, the solution is:
\[
k = -4, \quad l = 1
\]

### Final Answer:
The integers \( k \) and \( l \) that satisfy \( \gcd(3, 13) = 3k + 13l \) are:
\[
k = -4, \quad l = 1
\]

Do you want more details or have any questions? 

Here are 5 related questions to explore:

1. How does the Euclidean algorithm work for larger numbers?
2. Can you prove that the gcd is the smallest linear combination of \( a \) and \( b \)?
3. What is the significance of the Extended Euclidean Algorithm in number theory?
4. How would you solve for \( k \) and \( l \) if \( a \) and \( b \) were negative?
5. Can you apply this algorithm to solve Diophantine equations?

**Tip:** The Extended Euclidean Algorithm is very useful for solving Diophantine equations and finding modular inverses in number theory.

Learn how to find integers k and l such that gcd(3, 13) = 3k + 13l using the Extended Euclidean Algorithm. This problem explores key concepts in number theory, including the Euclidean Algorithm and its extended version. Suitable for high school students in Grades 9-12.

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