Stelle folgende Zahlen als Produkt von Primzahlpotenzen dar.

  1. ´a = 38!´
  2. ´b = 5 039 994 960´
  3. ´c = 2 441 348 099´
Hint 1
Zu 3: Bilde ´sqrt(c)´

Approach

Teil 1

´38! = 2^(e_1) * 3^(e_2) * 5^(e_3) * 7^(e_4) * 11^(e_5) * 13^(e_6) * 17^(e_7) * 19^(e_8) * 23^(e_9) * 29^(e_10) * 31^(e_11) * 37^(e_12)´

´e_1 = |__38/2__| + |__38/4__| + |__38/8__| + |__38/16__| + |__38/32__| = 19 + 9 + 4 + 2 + 1 = 35´

´e_2 = |__38/3__| + |__38/9__| + |__38/27__| = 12 + 4 + 1 = 17´

´e_3 = |__38/5__| + |__38/25__| = 9´

´e_4 = |__38/7__| = 5´

´e_5 = |__38/11__| = 3´

´e_6 = |__38/13__| = 2´

´e_7 = |__38/17__| = 2´

´e_8 = |__38/19__| = 2´

´e_9 = |__38/23__| = 1´

´e_10 = |__38/29__| = 1´

´e_11 = |__38/31__| = 1´

´e_12 = |__38/37__| = 1´

´38! = 2^35 * 3^17 * 5^9 * 7^5 * 11^3 * 13^2 * 17^2 * 19^2 * 23 * 29 * 31 * 37´

Teil 2

´5039994960 / 2 = 2519997480´

´… / 2 = 1259998740´

´… / 2 = 629999370´

´… / 2 = 314999685´

´… / 3 = 104999895´

´… / 3 = 34999965´

´… / 3 = 11666655´

´… / 3 = 3888885´

´… / 3 = 1296295´

´… / 5 = 259259´

´… / 7 = 37037´

´… / 7 = 5291´

´… / 11 = 481´

´… / 13 = 37´

´5039994960 = 2^4 * 3^5 * 5 * 7^2 * 11 * 13 * 37´

Teil 3

´sqrt(2441348099) ~~ 49410´ ´c = 49409 * 49411´


Solution
    1. ´38! = 2^35 * 3^17 * 5^9 * 7^5 * 11^3 * 13^2 * 17^2 * 19^2 * 23 * 29 * 31 * 37´
    2. ´5039994960 = 2^4 * 3^5 * 5 * 7^2 * 11 * 13 * 37´
    3. ´2441348099 = 49409 * 49411´
  • URL:
  • Language:
  • Subjects: math
  • Type: Transform
  • Duration: 30min
  • Credits: 3
  • Difficulty: 0.4
  • Tags: hpi prime prime power
  • Note:
    HPI, 2014-04-07, Mathe 2, Blatt 1, Aufgabe 4
  • Created By: adius
  • Created At:
    2014-07-25 20:25:03 UTC
  • Last Modified:
    2014-07-25 20:25:28 UTC