We will solve this problem in two parts:

1. Finding the G.C.D. (Greatest Common Divisor) of 5678 and 1234 using the Euclidean algorithm.
2. Finding integers $s$ and $t$ such that $\text{G.C.D.}(5678, 1234) = 5678s + 1234t$ using the Extended Euclidean algorithm.

Step 1: Euclidean Algorithm to find G.C.D.

We perform the Euclidean algorithm by repeatedly applying the division algorithm:
$a = bq + r$ where $a$ and $b$ are integers, and $q$ is the quotient and $r$ is the remainder.

Let’s apply this to 5678 and 1234.

$5678 \div 1234 = 4 \quad \text{with remainder} \quad r = 5678 - 4 \times 1234 = 5678 - 4936 = 742$ Now, apply the algorithm to 1234 and 742:

$1234 \div 742 = 1 \quad \text{with remainder} \quad r = 1234 - 1 \times 742 = 1234 - 742 = 492$ Next, apply the algorithm to 742 and 492:

$742 \div 492 = 1 \quad \text{with remainder} \quad r = 742 - 1 \times 492 = 742 - 492 = 250$ Next, apply the algorithm to 492 and 250:

$492 \div 250 = 1 \quad \text{with remainder} \quad r = 492 - 1 \times 250 = 492 - 250 = 242$ Next, apply the algorithm to 250 and 242:

$250 \div 242 = 1 \quad \text{with remainder} \quad r = 250 - 1 \times 242 = 250 - 242 = 8$ Next, apply the algorithm to 242 and 8:

$242 \div 8 = 30 \quad \text{with remainder} \quad r = 242 - 30 \times 8 = 242 - 240 = 2$ Finally, apply the algorithm to 8 and 2:

$8 \div 2 = 4 \quad \text{with remainder} \quad r = 8 - 4 \times 2 = 0$ Since the remainder is 0, the G.C.D. is 2.

Step 2: Extended Euclidean Algorithm to find $s$ and $t$

Now, we need to find integers $s$ and $t$ such that:

$\text{G.C.D.}(5678, 1234) = 5678s + 1234t = 2$

We work backwards from the Euclidean steps:

$2 = 242 - 30 \times 8$ Substitute $8 = 250 - 1 \times 242$:

$2 = 242 - 30 \times (250 - 1 \times 242) = 242 - 30 \times 250 + 30 \times 242 = 31 \times 242 - 30 \times 250$ Substitute $242 = 492 - 1 \times 250$:

$2 = 31 \times (492 - 1 \times 250) - 30 \times 250 = 31 \times 492 - 31 \times 250 - 30 \times 250 = 31 \times 492 - 61 \times 250$ Substitute $250 = 742 - 1 \times 492$:

$2 = 31 \times 492 - 61 \times (742 - 1 \times 492) = 31 \times 492 - 61 \times 742 + 61 \times 492 = 92 \times 492 - 61 \times 742$ Substitute $492 = 1234 - 1 \times 742$:

$2 = 92 \times (1234 - 1 \times 742) - 61 \times 742 = 92 \times 1234 - 92 \times 742 - 61 \times 742 = 92 \times 1234 - 153 \times 742$ Substitute $742 = 5678 - 4 \times 1234$:

$2 = 92 \times 1234 - 153 \times (5678 - 4 \times 1234) = 92 \times 1234 - 153 \times 5678 + 612 \times 1234$ Simplifying:

$2 = (92 + 612) \times 1234 - 153 \times 5678 = 704 \times 1234 - 153 \times 5678$

Thus, the integers $s$ and $t$ are:

$s = -153, \quad t = 704$

Final Answer:

• G.C.D.(5678, 1234) = 2
• Integers $s = -153$ and $t = 704$ satisfy $5678s + 1234t = 2$.

Would you like more details or have any questions?

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Relative Questions:

1. How is the Euclidean algorithm related to finding the G.C.D.?
2. What is the purpose of the Extended Euclidean Algorithm?
3. Can the G.C.D. of two numbers be greater than the smaller of the two numbers?
4. How does the G.C.D. help in solving Diophantine equations?
5. How would you apply the Euclidean algorithm to three or more numbers?

Tip: The Extended Euclidean Algorithm not only finds the G.C.D. but also provides a way to express it as a linear combination of the original numbers.