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Properties of the GCD Theorem and Application in Euclidean Algorithm

Find integers x, y such that xa + yb = gcd(a, b) using Euclidean Algorithm. Example of gcd(78, 216) = 6. Proof for gcd(a, b) = 1 when a, b are relatively prime. Proof that if a | bc and a, b are coprime, then a | c.

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Properties of the GCD Theorem and Application in Euclidean Algorithm

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  1. Properties of the gcd Theorem: For any two integers a,b there exist integers x,y such that xa + yb = gcd(a,b). A proof will not be given at this point. However, x,y can be found by applying the Euclidean algorithm to a and b, and then, working from the end and for every division of u by v with quotient q and positive reminder r such that u = qv + r, replacing r by u - qv.

  2. The gcd Example: Applying the Euclidean algorithm to find the gcd of 78 and 216:  (78) (216) (216) = 2(78) + (60) I (60) (78) (78) = 1(60) + (18) II (18) (60) (60) = 3(18) + (6) III (6) (18) (18) = 3(6) + (0) IV 

  3. The gcd gcd(78,216)=6 The application of the Euclidean algorithm is completed.

  4. The gcd Finding x, y such that xa + yb = gcd(a,b): By III: (6) = (60) - 3(18) By II: (18) = (78) - 1(60), so: (60) - 3(18) = (60) - 3[(78) - 1(60)] = 4(60) - 3(78) By I: (60) = (216) - 2(78), so: 4(60) - 3(78) = 4[(216) - 2(78)] - 3(78) = 4(216) - 11(78)  Conclusion: (6) = 4(216) - 11(78)

  5. The gcd Theorem: Let a, b be relatively prime integers. Then there exists an integer x such that xa  1 (mod b). Proof: By the previous theorem, there exist integers x,y such that xa + yb = gcd(a,b). Since a,b are relatively prime, gcd(a,b)=1. So xa + yb = 1, hence xa = -yb +1, so xa  1 (mod b).

  6. The gcd Theorem: Let a,b,c be integers such that a,b are relatively prime and a | bc. Then a | c. Proof: By a previous theorem there exist integers x,y such that xa + yb = gcd(a,b). Since gcd(x,y)=1, xa + yb = 1.

  7. The gcd Since a | bc, there exists an integer m such that bc = ma. So c = c(xa + yb) = xac + ybc = xac + yma = a(xc + ym). So a divides c.

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