Texts: Use the extended Euclidean algorithm to find the greatest common divisor of 9,864 and 1,134 and express it as a linear combination of 9,864 and 1,134.
Step 1: Find q1 and r1, so that
9,864 = 1,134q1 + r1, where 0 ≤ r1 < 1,134. Then r1 = 9,864 - 1,134q1.
Step 2: Find q2 and r2 so that
1,134 = r1q2 + r2, where 0 ≤ r2 < r1. Then r2 = 1,134 - r1q2.
Step 3: Find q3 and r3 so that
r1 = r2q3 + r3, where 0 ≤ r3 < r2. Then r3 = r1 - r2q3.
Step 4: Find q4 and r4 so that
r2 = r3q4 + r4, where 0 ≤ r4 < r3. Then r4 = r2 - r3q4.
Step 5: Find q5 and r5 so that
r3 = r4q5 + r5, where 0 ≤ r5 < r4. Then r5 = r3 - r4q5.
Step 6: Conclude that gcd (9,864, 1,134) equals which of the following:
a) gcd(9,864, 1,134) = r4 - r5q3
b) gcd(9,864, 1,134) = r3 - r4q4
c) gcd(9,864, 1,134) = r1 - r3q5
d) gcd(9,864, 1,134) = r1 - r4q5
Conclusion: Substitute numerical values backward through the preceding steps, simplifying the results for each step, until you have found numbers s and t so that
gcd (9,864, 1,134) = 9,864s + 1,134t,
where s =
and t =