Express the greatest common divisor of each of these pairs of integers a, b as sa + t b, for some
integers s and t 3454 4666

We are tasked with finding the greatest common divisor (gcd) of 3454 and 4666, and expressing it as a linear combination of 3454 and 4666, i.e., in the form:

\[
\gcd(3454, 4666) = s \cdot 3454 + t \cdot 4666
\]

### Step 1: Find the gcd using the Euclidean Algorithm
The Euclidean algorithm helps us find the gcd by performing repeated division and taking remainders.

1. Divide 4666 by 3454:
   \[
   4666 \div 3454 = 1 \quad \text{remainder} = 1212
   \]
   So, \( 4666 = 1 \cdot 3454 + 1212 \).

2. Divide 3454 by 1212:
   \[
   3454 \div 1212 = 2 \quad \text{remainder} = 1030
   \]
   So, \( 3454 = 2 \cdot 1212 + 1030 \).

3. Divide 1212 by 1030:
   \[
   1212 \div 1030 = 1 \quad \text{remainder} = 182
   \]
   So, \( 1212 = 1 \cdot 1030 + 182 \).

4. Divide 1030 by 182:
   \[
   1030 \div 182 = 5 \quad \text{remainder} = 100
   \]
   So, \( 1030 = 5 \cdot 182 + 100 \).

5. Divide 182 by 100:
   \[
   182 \div 100 = 1 \quad \text{remainder} = 82
   \]
   So, \( 182 = 1 \cdot 100 + 82 \).

6. Divide 100 by 82:
   \[
   100 \div 82 = 1 \quad \text{remainder} = 18
   \]
   So, \( 100 = 1 \cdot 82 + 18 \).

7. Divide 82 by 18:
   \[
   82 \div 18 = 4 \quad \text{remainder} = 10
   \]
   So, \( 82 = 4 \cdot 18 + 10 \).

8. Divide 18 by 10:
   \[
   18 \div 10 = 1 \quad \text{remainder} = 8
   \]
   So, \( 18 = 1 \cdot 10 + 8 \).

9. Divide 10 by 8:
   \[
   10 \div 8 = 1 \quad \text{remainder} = 2
   \]
   So, \( 10 = 1 \cdot 8 + 2 \).

10. Divide 8 by 2:
    \[
    8 \div 2 = 4 \quad \text{remainder} = 0
    \]
    So, the gcd is \( \gcd(3454, 4666) = 2 \).

### Step 2: Back-substitute to find \( s \) and \( t \)
Now, we backtrack to express the gcd as a linear combination of 3454 and 4666.

1. \( 2 = 10 - 1 \cdot 8 \)
   
2. \( 2 = 10 - 1 \cdot (18 - 1 \cdot 10) = 2 \cdot 10 - 1 \cdot 18 \)

3. \( 2 = 2 \cdot (100 - 1 \cdot 82) - 1 \cdot 18 = 2 \cdot 100 - 2 \cdot 82 - 1 \cdot 18 \)

4. \( 2 = 2 \cdot 100 - 2 \cdot (182 - 1 \cdot 100) - 1 \cdot 18 = 4 \cdot 100 - 2 \cdot 182 - 1 \cdot 18 \)

5. \( 2 = 4 \cdot (1030 - 5 \cdot 182) - 2 \cdot 182 - 1 \cdot 18 = 4 \cdot 1030 - 22 \cdot 182 - 1 \cdot 18 \)

6. \( 2 = 4 \cdot 1030 - 22 \cdot (1212 - 1 \cdot 1030) - 1 \cdot 18 = 26 \cdot 1030 - 22 \cdot 1212 - 1 \cdot 18 \)

7. \( 2 = 26 \cdot (3454 - 2 \cdot 1212) - 22 \cdot 1212 - 1 \cdot 18 = 26 \cdot 3454 - 74 \cdot 1212 - 1 \cdot 18 \)

8. \( 2 = 26 \cdot 3454 - 74 \cdot (4666 - 1 \cdot 3454) - 1 \cdot 18 = 100 \cdot 3454 - 74 \cdot 4666 - 1 \cdot 18 \)

Thus, we have expressed \( \gcd(3454, 4666) = 2 \) as:

\[
2 = 100 \cdot 3454 - 74 \cdot 4666
\]

So, \( s = 100 \) and \( t = -74 \).

---

Would you like more details or have any questions? Here are some related questions:

1. How does the Euclidean algorithm work in finding gcds for any two integers?
2. Can you explain the back-substitution process in more detail?
3. What are the applications of expressing the gcd as a linear combination?
4. How can this method be generalized for other pairs of numbers?
5. Why is the gcd always the smallest positive linear combination of the two numbers?

**Tip**: The gcd of two numbers is also the largest integer that divides both numbers exactly.

Express the greatest common divisor of each of these pairs of integers a, b as sa + t b, for some integers s and t. 3454, 4666.

Learn how to find the greatest common divisor (GCD) of 3454 and 4666 using the Euclidean Algorithm, and express it as a linear combination of the two numbers. This detailed step-by-step solution involves number theory concepts like Bézout's identity. Suitable for Grades 10-12.

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