Probeklausur_2019-12-17.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(999999999.9090909, 0.0)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1000**3 - 1 / 11, 0/11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pow(1000, 19**19, 11) % 11"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4, [1, 161, 649, 809])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = []\n",
    "for i in range(1, 811):\n",
    "    if (i**2)%810 == 1:\n",
    "        res.append((i))\n",
    "            \n",
    "len(res), res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def find_inv(z):\n",
    "    res = []\n",
    "    for x in range(-z, z + 1):\n",
    "        for y in range(-z, z + 1):\n",
    "            if (x * y) % z == 1:\n",
    "                res.append((x, y, x*y%z))\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-35, -35, 1),\n",
       " (-35, 1, 1),\n",
       " (-31, -7, 1),\n",
       " (-31, 29, 1),\n",
       " (-29, -5, 1),\n",
       " (-29, 31, 1),\n",
       " (-25, -13, 1),\n",
       " (-25, 23, 1),\n",
       " (-23, -11, 1),\n",
       " (-23, 25, 1),\n",
       " (-19, -19, 1),\n",
       " (-19, 17, 1),\n",
       " (-17, -17, 1),\n",
       " (-17, 19, 1),\n",
       " (-13, -25, 1),\n",
       " (-13, 11, 1),\n",
       " (-11, -23, 1),\n",
       " (-11, 13, 1),\n",
       " (-7, -31, 1),\n",
       " (-7, 5, 1),\n",
       " (-5, -29, 1),\n",
       " (-5, 7, 1),\n",
       " (-1, -1, 1),\n",
       " (-1, 35, 1),\n",
       " (1, -35, 1),\n",
       " (1, 1, 1),\n",
       " (5, -7, 1),\n",
       " (5, 29, 1),\n",
       " (7, -5, 1),\n",
       " (7, 31, 1),\n",
       " (11, -13, 1),\n",
       " (11, 23, 1),\n",
       " (13, -11, 1),\n",
       " (13, 25, 1),\n",
       " (17, -19, 1),\n",
       " (17, 17, 1),\n",
       " (19, -17, 1),\n",
       " (19, 19, 1),\n",
       " (23, -25, 1),\n",
       " (23, 11, 1),\n",
       " (25, -23, 1),\n",
       " (25, 13, 1),\n",
       " (29, -31, 1),\n",
       " (29, 5, 1),\n",
       " (31, -29, 1),\n",
       " (31, 7, 1),\n",
       " (35, -1, 1),\n",
       " (35, 35, 1)]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "find_inv(36)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "29\n",
      "31\n",
      "23\n",
      "25\n",
      "17\n",
      "19\n",
      "11\n",
      "13\n",
      "5\n",
      "7\n",
      "35\n"
     ]
    }
   ],
   "source": [
    "from sympy import mod_inverse\n",
    "for x in range(1, 36):\n",
    "    try:\n",
    "        print(mod_inverse(x, 36))\n",
    "    except:\n",
    "        pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71] 73\n"
     ]
    }
   ],
   "source": [
    "from sympy import prime\n",
    "\n",
    "# prime(q) muss 1 größer als 20 sein\n",
    "print([prime(i) for i in range(1, 21)], prime(21))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5940.0"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import factorial\n",
    "\n",
    "3**3 * factorial(12) / (factorial(3) * factorial(12 - 3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.ntheory.modular import crt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a=31 -> (mpz(591), 1085)\n",
      "a=32 -> (mpz(66), 1120)\n",
      "a=33 -> (mpz(101), 1155)\n",
      "a=34 -> (mpz(206), 1190)\n",
      "a=35 -> None\n",
      "a=36 -> (mpz(1046), 1260)\n",
      "a=37 -> (mpz(1186), 1295)\n",
      "a=38 -> (mpz(1256), 1330)\n",
      "a=39 -> (mpz(626), 1365)\n"
     ]
    }
   ],
   "source": [
    "for a in range(31, 40):\n",
    "    c = crt([5, a, 7], [1, 2, 3])\n",
    "    print('a={} -> {}'.format(a, c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Fraction(1, 25),\n",
       " 3.1622776601683795,\n",
       " 3.3166247903554,\n",
       " 11.0,\n",
       " 0.0625,\n",
       " 0.0949157995752499)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from fractions import Fraction\n",
    "from math import sqrt\n",
    "\n",
    "Fraction(3, 75), sqrt(sqrt(100)), sqrt(sqrt(121)), sqrt(121), float(Fraction(1, 16)), sqrt(111)/111"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.04"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.04 * 25"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "215 --> 3335000\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "215"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = 1\n",
    "\n",
    "while True:\n",
    "    res = 0\n",
    "    \n",
    "    for i in range(42, b + 1):\n",
    "        res += (i**2) + (i) + 3\n",
    "    \n",
    "    if res == 3335000:\n",
    "        print('{} --> {}'.format(b, res))\n",
    "        break\n",
    "    b += 1\n",
    "b\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  1379,\n",
       "  9571,\n",
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       "  9781,\n",
       "  9783,\n",
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       "  1597,\n",
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       "  1635,\n",
       "  9827,\n",
       "  1637,\n",
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       "  9831,\n",
       "  1643,\n",
       "  9835,\n",
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       "  1653,\n",
       "  9845,\n",
       "  9847,\n",
       "  1657,\n",
       "  1659,\n",
       "  9851,\n",
       "  9853,\n",
       "  9857,\n",
       "  9861,\n",
       "  9863,\n",
       "  1673,\n",
       "  9865,\n",
       "  1675,\n",
       "  9867,\n",
       "  1679,\n",
       "  9871,\n",
       "  9873,\n",
       "  1683,\n",
       "  9875,\n",
       "  1685,\n",
       "  1687,\n",
       "  1689,\n",
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       "  1705,\n",
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       "  1725,\n",
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       "  1745,\n",
       "  1749,\n",
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       "  1759,\n",
       "  1763,\n",
       "  1765,\n",
       "  1769,\n",
       "  1783,\n",
       "  1785,\n",
       "  1789,\n",
       "  1793,\n",
       "  1795,\n",
       "  1803,\n",
       "  1805,\n",
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       "  1823,\n",
       "  1825,\n",
       "  1827,\n",
       "  1829,\n",
       "  1835,\n",
       "  1837,\n",
       "  1839,\n",
       "  1843,\n",
       "  1845,\n",
       "  1847,\n",
       "  1849,\n",
       "  1853,\n",
       "  1857,\n",
       "  1859,\n",
       "  1863,\n",
       "  1865,\n",
       "  1867,\n",
       "  1869,\n",
       "  1873,\n",
       "  1875,\n",
       "  1879,\n",
       "  1893,\n",
       "  1895,\n",
       "  1897,\n",
       "  1903,\n",
       "  1905,\n",
       "  1907,\n",
       "  1923,\n",
       "  1925,\n",
       "  1927,\n",
       "  1935,\n",
       "  1937,\n",
       "  1943,\n",
       "  1945,\n",
       "  1947,\n",
       "  1953,\n",
       "  1957,\n",
       "  1963,\n",
       "  1965,\n",
       "  1967,\n",
       "  1973,\n",
       "  1975,\n",
       "  1983,\n",
       "  1985,\n",
       "  1987,\n",
       "  2013,\n",
       "  2015,\n",
       "  2017,\n",
       "  2019,\n",
       "  2031,\n",
       "  2035,\n",
       "  2037,\n",
       "  2039,\n",
       "  2041,\n",
       "  2043,\n",
       "  2045,\n",
       "  2047,\n",
       "  2049,\n",
       "  2051,\n",
       "  2053,\n",
       "  2057,\n",
       "  2059,\n",
       "  2061,\n",
       "  2063,\n",
       "  2065,\n",
       "  2067,\n",
       "  2069,\n",
       "  2071,\n",
       "  2073,\n",
       "  2075,\n",
       "  2079,\n",
       "  2081,\n",
       "  2083,\n",
       "  2085,\n",
       "  2087,\n",
       "  2089,\n",
       "  2091,\n",
       "  2093,\n",
       "  2095,\n",
       "  2097,\n",
       "  2103,\n",
       "  2105,\n",
       "  2107,\n",
       "  2109,\n",
       "  2135,\n",
       "  2137,\n",
       "  2139,\n",
       "  2143,\n",
       "  2145,\n",
       "  2147,\n",
       "  2149,\n",
       "  2153,\n",
       "  2157,\n",
       "  2159,\n",
       "  2163,\n",
       "  2165,\n",
       "  2167,\n",
       "  2169,\n",
       "  2173,\n",
       "  2175,\n",
       "  2179,\n",
       "  2183,\n",
       "  2185,\n",
       "  2187,\n",
       "  2189,\n",
       "  2193,\n",
       "  2195,\n",
       "  2197,\n",
       "  2301,\n",
       "  2305,\n",
       "  2307,\n",
       "  2309,\n",
       "  2315,\n",
       "  2317,\n",
       "  2319,\n",
       "  2341,\n",
       "  2345,\n",
       "  2347,\n",
       "  2349,\n",
       "  2351,\n",
       "  2357,\n",
       "  2359,\n",
       "  2361,\n",
       "  2365,\n",
       "  2367,\n",
       "  2369,\n",
       "  2371,\n",
       "  2375,\n",
       "  2379,\n",
       "  2381,\n",
       "  2385,\n",
       "  2387,\n",
       "  2389,\n",
       "  2391,\n",
       "  2395,\n",
       "  2397,\n",
       "  2401,\n",
       "  2403,\n",
       "  2405,\n",
       "  2407,\n",
       "  2409,\n",
       "  2413,\n",
       "  2415,\n",
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       "  2419,\n",
       "  2431,\n",
       "  2435,\n",
       "  2437,\n",
       "  2439,\n",
       "  2451,\n",
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       "  2457,\n",
       "  2459,\n",
       "  2461,\n",
       "  2463,\n",
       "  2465,\n",
       "  2467,\n",
       "  2469,\n",
       "  2471,\n",
       "  2473,\n",
       "  2475,\n",
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       "  2481,\n",
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       "  2485,\n",
       "  2487,\n",
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       "  2503,\n",
       "  2507,\n",
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       "  2519,\n",
       "  2531,\n",
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       "  2539,\n",
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       "  2549,\n",
       "  2561,\n",
       "  2563,\n",
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       "  2571,\n",
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       "  2581,\n",
       "  2583,\n",
       "  2587,\n",
       "  2589,\n",
       "  2591,\n",
       "  2593,\n",
       "  2597,\n",
       "  2601,\n",
       "  2603,\n",
       "  2605,\n",
       "  2607,\n",
       "  2609,\n",
       "  2613,\n",
       "  2615,\n",
       "  2617,\n",
       "  2619,\n",
       "  2631,\n",
       "  2635,\n",
       "  2637,\n",
       "  2639,\n",
       "  2641,\n",
       "  2643,\n",
       "  2645,\n",
       "  2647,\n",
       "  2649,\n",
       "  2651,\n",
       "  2653,\n",
       "  2657,\n",
       "  2659,\n",
       "  2671,\n",
       "  2673,\n",
       "  2675,\n",
       "  2679,\n",
       "  2681,\n",
       "  2683,\n",
       "  2685,\n",
       "  2687,\n",
       "  2689,\n",
       "  2691,\n",
       "  2693,\n",
       "  2695,\n",
       "  2697,\n",
       "  2701,\n",
       "  2703,\n",
       "  2705,\n",
       "  2709,\n",
       "  2713,\n",
       "  2715,\n",
       "  2719,\n",
       "  2731,\n",
       "  2735,\n",
       "  2739,\n",
       "  2741,\n",
       "  2743,\n",
       "  2745,\n",
       "  2749,\n",
       "  2751,\n",
       "  2753,\n",
       "  2759,\n",
       "  2761,\n",
       "  2763,\n",
       "  2765,\n",
       "  2769,\n",
       "  2781,\n",
       "  2783,\n",
       "  2785,\n",
       "  2789,\n",
       "  2791,\n",
       "  2793,\n",
       "  2795,\n",
       "  2801,\n",
       "  2803,\n",
       "  2805,\n",
       "  2807,\n",
       "  2809,\n",
       "  2813,\n",
       "  2815,\n",
       "  2817,\n",
       "  2819,\n",
       "  2831,\n",
       "  2835,\n",
       "  2837,\n",
       "  2839,\n",
       "  2841,\n",
       "  2843,\n",
       "  2845,\n",
       "  2847,\n",
       "  2849,\n",
       "  2851,\n",
       "  2853,\n",
       "  2857,\n",
       "  2859,\n",
       "  2861,\n",
       "  2863,\n",
       "  2865,\n",
       "  2867,\n",
       "  2869,\n",
       "  2871,\n",
       "  2873,\n",
       "  2875,\n",
       "  2879,\n",
       "  2891,\n",
       "  2893,\n",
       "  2895,\n",
       "  2897,\n",
       "  2901,\n",
       "  2903,\n",
       "  2905,\n",
       "  2907,\n",
       "  2913,\n",
       "  2915,\n",
       "  2917,\n",
       "  2931,\n",
       "  2935,\n",
       "  2937,\n",
       "  2941,\n",
       "  2943,\n",
       "  2945,\n",
       "  2947,\n",
       "  2951,\n",
       "  2953,\n",
       "  2957,\n",
       "  2961,\n",
       "  2963,\n",
       "  2965,\n",
       "  2967,\n",
       "  2971,\n",
       "  2973,\n",
       "  2975,\n",
       "  2981,\n",
       "  2983,\n",
       "  2985,\n",
       "  2987,\n",
       "  3015,\n",
       "  3017,\n",
       "  3019,\n",
       "  3021,\n",
       "  3025,\n",
       "  3027,\n",
       "  3029,\n",
       "  3041,\n",
       "  3045,\n",
       "  3047,\n",
       "  3049,\n",
       "  3051,\n",
       "  3057,\n",
       "  3059,\n",
       "  3061,\n",
       "  3065,\n",
       "  3067,\n",
       "  3069,\n",
       "  3071,\n",
       "  3075,\n",
       "  3079,\n",
       "  3081,\n",
       "  3085,\n",
       "  3087,\n",
       "  3089,\n",
       "  3091,\n",
       "  3095,\n",
       "  3097,\n",
       "  3105,\n",
       "  3107,\n",
       "  3109,\n",
       "  3125,\n",
       "  3127,\n",
       "  3129,\n",
       "  3145,\n",
       "  3147,\n",
       "  3149,\n",
       "  3157,\n",
       "  3159,\n",
       "  3165,\n",
       "  3167,\n",
       "  3169,\n",
       "  3175,\n",
       "  3179,\n",
       "  3185,\n",
       "  3187,\n",
       "  3189,\n",
       "  3195,\n",
       "  3197,\n",
       "  3201,\n",
       "  3205,\n",
       "  3207,\n",
       "  ...})"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = { x1*1000 + x2*100 + x3*10 + x4 \n",
    "        for x1 in range(1, 10)\n",
    "        for x2 in range(0, 10)\n",
    "        for x3 in range(0, 10)\n",
    "        for x4 in [1, 3, 5, 7, 9]\n",
    "        if x1 != x2 and x1 != x3 and x1 != x4 and x2 != x3 and x2 != x4 and x3 != x4\n",
    "}\n",
    "\n",
    "len(res), res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "13\n"
     ]
    }
   ],
   "source": [
    "for i in range(29):\n",
    "    if (i*9) % 29 == 1:\n",
    "        print(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 7, 7)"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(13 * 9) % 29, (5 * 13) % 29, pow(7, 1000000, 29)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.43, 0.480740698407860230965967436087996657705204)"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import N, sqrt\n",
    "\n",
    "N(sqrt(71) - 8, 2), N(sqrt(42) - 6, 42) "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
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    "version": 3
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