Finde alle ´x in ZZ´ für die gleichzeitig folgende Kongruenzen gelten:

  1. ´x -= 1 (mod 5)´
  2. ´x -= 2 (mod 11)´
  3. ´x -= 3 (mod 9)´
Approach

´x -= 1 (mod 5)´ ´<=> x - 1 in 5 * ZZ´ ´<=> x = y5 + 1 (y in ZZ)´

In 2: ´1 + 5y -= 2 (mod 11)´ | -1 ´5y -= 1 (mod 11)´ ´y -= 9 (mod 11)´

´=> y = 9 + 11z (z in ZZ)´

In 1: ´x = 5(9 + 11z) + 1 = 46 + 55z´

In 3: ´46 + 55z -= 3 (mod 9)´ ´1 + z -= 3 (mod 9)´ ´z -= 2 (mod 9)´

´=> z = 2 + 9w (w in ZZ)´

In 2: ´x = 46 + 55(2 + 9w) = 156 + 495w´

´x -= 156 (mod 495)´


Solution

  • ´x -= 156 (mod 495)´

  • URL:
  • Language:
  • Subjects: math
  • Type: Calculate
  • Duration: 20min
  • Credits: 3
  • Difficulty: 0.6
  • Tags: hpi congruence modulo
  • Note:
    HPI, 2014-06-10, Mathe 2, Aufgabe 40
  • Created By: adius
  • Created At:
    2014-07-26 18:06:23 UTC
  • Last Modified:
    2014-07-26 18:06:23 UTC