The linear combination of gcd(252, 198) = 18 is ________.

Step 1 (Euclidean Division Steps): We find the greatest common divisor of $252$ and $198$ using the Euclidean Algorithm:

1. $252=198\times 1+54\text{— (Equation 1)}$

2. $198=54\times 3+36\text{— (Equation 2)}$

3. $54=36\times 1+18\text{— (Equation 3)}$

4. $36=18\times 2+0$

So $\gcd (252,198)=18$.

Step 2 (Back-substitution): We express 18 as a linear combination of 252 and 198 by working backwards:

• From Equation 3:

• From Equation 2: $36=198-54\times 3$. Substitute this into the equation:

• From Equation 1: $54=252-198\times 1$. Substitute this into the equation:

Step 3 (Conclusion): A linear combination representing the GCD is 252(4) + 198(-5).