To find the coefficients \(s\) and \(t\) in the linear combination \(6 = 2322s + 654t\), we will use the results from the Euclidean algorithm applied to \(2322\) and \(654\). Here, we will backtrack through the steps to express 6 as a linear combination of \(2322\) and \(654\).

### Back Substitution Steps:

We already know from the previous solution:
\[
6 = 20 \times 2322 - 71 \times 654
\]
So, the values of \(s\) and \(t\) are:
\[
s = 20 \quad \text{and} \quad t = -71
\]

### Verification:

Substitute \(s = 20\) and \(t = -71\) back into the original equation to verify:
\[
6 = 2322 \times 20 + 654 \times (-71)
\]
Calculate each term:
\[
2322 \times 20 = 46440
\]
\[
654 \times (-71) = -46434
\]
Add these together:
\[
46440 - 46434 = 6
\]
The calculation is correct, so \(s = 20\) and \(t = -71\) are indeed the correct coefficients.

Would you like any further details or have any more questions?

Here are some related questions you might find useful:
1. How would you find the coefficients \(s\) and \(t\) if the GCD was different from 6?
2. Can the values of \(s\) and \(t\) be negative in general?
3. What is the importance of finding such coefficients in number theory?
4. How does the method change if \(2322\) and \(654\) were swapped?
5. Can these coefficients \(s\) and \(t\) be found using matrix methods?

**Tip:** The coefficients \(s\) and \(t\) can be interpreted as the weights needed to combine two numbers to reach the GCD. This concept is especially useful in modular arithmetic and cryptography!

Learn how to solve the equation 6 = 2322s + 654t to find the coefficients s and t using Number Theory and the Extended Euclidean Algorithm. This problem involves solving a linear Diophantine equation and applying Bezout's Identity. Suitable for advanced high school students.

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