Abstract Algebra Theory and Applications - Computer Science ...

28 CHAPTER 1 THE INTEGERSThe Euclidean AlgorithmAmong other things, Theorem 1.4 allows us to compute the greatest commondivisor of two integers.Example 4. Let us compute the greatest common divisor of 945 and 2415.First observe that2415 = 945 · 2 + 525945 = 525 · 1 + 420525 = 420 · 1 + 105420 = 105 · 4 + 0.Reversing our steps, 105 divides 420, 105 divides 525, 105 divides 945, and105 divides 2415. Hence, 105 divides both 945 and 2415. If d were anothercommon divisor of 945 and 2415, then d would also have to divide 105.Therefore, gcd(945, 2415) = 105.If we work backward through the above sequence of equations, we canalso obtain numbers r and s such that 945r + 2415s = 105. Observe that105 = 525 + (−1) · 420= 525 + (−1) · [945 + (−1) · 525]= 2 · 525 + (−1) · 945= 2 · [2415 + (−2) · 945] + (−1) · 945= 2 · 2415 + (−5) · 945.So r = −5 and s = 2. Notice that r and s are not unique, since r = 41 ands = −16 would also work.To compute gcd(a, b) = d, we are using repeated divisions to obtain adecreasing sequence of positive integers r 1 > r 2 > · · · > r n = d; that is,b = aq 1 + r 1a = r 1 q 2 + r 2r 1 = r 2 q 3 + r 3.r n−2 = r n−1 q n + r nr n−1 = r n q n+1 .