Es sei ´a = 8897´, ´b = 13237´, ´c = 17507´. Bestimme den ´gcd(a, b, c)´ und ´lcm(a, b, c)´ und stelle ´gcd(a, b, c)´ in der Form ´xa + yb + zc´ mit ´x, y, z´ in ´ZZ´ dar.

Approach

"´13237 = 1 \* 8897 + 4340´\n´8897 = 2 \* 4340 + 217´\n´4340 = 20 \* 217 + 0´\n\n´gcd(13237, 8897) = 217´\n´lcm(13237, 8897) = (13237 \* 8897)/gcd(13237, 8897) = 117769589/217 = 542717´\n\n\n´gcd(a,b,c) = gcd(gcd(a,b),c)´\n\n\n´17507 = 80 \* 217 + 147´\n´217 = 1 \* 147 + 70´\n´147 = 2 \* 70 + 7´\n´70 = 10 \* 7 + 0´\n\n´gcd(gcd(13237, 8897), 17507) = 7´\n´lcm(lcm(13237, 8897), 17507) = 542717 \* 17507/gcd(542717, 17507) = (542717 \* 17507)/17507 = 542717´\n\nA:\n´147 = 17507 - 80 \* 217´\n´70 = 217 - 147 = 217 - (17507 - 80 \* 217) = 81 \* 217 - 17507´\n´7 = 147 - 2 \* 70 = 17507 - 80 \* 217 - 2(81 \* 217 - 17507) = 3 \* 17507 - 242 \* 217´\n\nB:\n´4340 = 13237 - 1 \* 8897´\n´217 = 8897 - 2 \* 4340 = 8897 - 2(13237 - 8897) = 3 \* 8897 - 2 \* 13237´\n\nB in A:\n\n´7 = 3 \* 17507 - 242(3 \* 8897 - 2 \* 13237) = 3 \* 17507 - 726 \* 8897 + 484 \* 13237´\n\n´gcd(13237, 8897, 17507) = -726 \* 8897 + 484 \* 13237 + 3 \* 17507´"


Solution

  • ´gcd(13237, 8897, 17507) = -726 * 8897 + 484 * 13237 + 3 * 17507 = 7´ ´lcm(13237, 8897, 17507) = 542717´

  • URL:
  • Language: Deutsch
  • Subjects: math
  • Type: Calculate
  • Duration: 25min
  • Credits: 3
  • Difficulty: 0.5
  • Tags: hpi gcd lcm
  • Note:
    HPI, 2014-04-14, Mathe 2, Aufgabe 7
  • Created By: ad-si
  • Created At:
    2014-07-25 22:27:02 UTC
  • Last Modified:
    2014-07-25 22:27:02 UTC