For each lair of integers find gcd(a,b) and integers k and l such that gcd(a,b)=ak+bl. A=62 and b=48

We are asked to find the greatest common divisor (GCD) of \( a = 62 \) and \( b = 48 \), and also find integers \( k \) and \( l \) such that:

\[
\gcd(a,b) = ak + bl
\]

### Step 1: Use the Euclidean Algorithm to find the GCD of 62 and 48.

The Euclidean algorithm involves repeated division and finding remainders:

1. \( 62 \div 48 = 1 \) with remainder \( 62 - 48 = 14 \), so:
   \[
   62 = 48 \times 1 + 14
   \]

2. \( 48 \div 14 = 3 \) with remainder \( 48 - 14 \times 3 = 6 \), so:
   \[
   48 = 14 \times 3 + 6
   \]

3. \( 14 \div 6 = 2 \) with remainder \( 14 - 6 \times 2 = 2 \), so:
   \[
   14 = 6 \times 2 + 2
   \]

4. \( 6 \div 2 = 3 \) with remainder 0, so:
   \[
   6 = 2 \times 3 + 0
   \]

Since the remainder is now 0, the GCD is the last non-zero remainder, which is \( 2 \).

Thus, \( \gcd(62, 48) = 2 \).

### Step 2: Use the Extended Euclidean Algorithm to express GCD as a linear combination.

We now work backwards from the Euclidean algorithm steps to express 2 as a combination of 62 and 48.

1. From \( 14 = 6 \times 2 + 2 \), we have:
   \[
   2 = 14 - 6 \times 2
   \]

2. From \( 6 = 48 - 14 \times 3 \), we have:
   \[
   2 = 14 - (48 - 14 \times 3) \times 2 = 14 - 2 \times 48 + 6 \times 14 = 7 \times 14 - 2 \times 48
   \]

3. From \( 14 = 62 - 48 \), we have:
   \[
   2 = 7 \times (62 - 48) - 2 \times 48 = 7 \times 62 - 7 \times 48 - 2 \times 48 = 7 \times 62 - 9 \times 48
   \]

Thus, \( 2 = 7 \times 62 - 9 \times 48 \).

### Step 3: Final Answer

The GCD is \( 2 \), and integers \( k \) and \( l \) are \( 7 \) and \( -9 \), respectively. So:

\[
\gcd(62, 48) = 62 \times 7 + 48 \times (-9)
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How does the Euclidean algorithm work in general?
2. Can the Extended Euclidean Algorithm be applied to larger integers?
3. What are the applications of finding integers \( k \) and \( l \) in cryptography?
4. How does finding the GCD relate to simplifying fractions?
5. Can the GCD of more than two numbers be found using a similar method?

**Tip:** The Extended Euclidean Algorithm is not only useful for finding GCDs but also for solving Diophantine equations, where integer solutions are needed.

For each pair of integers, find gcd(a,b) and integers k and l such that gcd(a,b)=ak+bl. A=62 and b=48

Learn how to find the greatest common divisor (GCD) of 62 and 48 using the Euclidean and Extended Euclidean algorithms. This detailed solution includes step-by-step instructions and finds integers k and l such that 62k + 48l = gcd(62, 48). Suitable for Grades 10-12.

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