{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Μέγιστος Κοινός Διαιρέτης και Ελάχιστο Κοινό Πολλαπλάσιο"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Έστω ένα ορθογώνιο διαστάσεων x=30cm, y=12cm. Να βρεθεί ο μικρότερος αριθμός από τετράγωνα (ακέραιας πλευράς) που το καλύπτει."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Απάντηση: ΜΚΔ(30,12) = 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# μια απλοϊκή υλοποίηση\n",
    "def gcd_naive(x, y):\n",
    "    z = min(x, y)\n",
    "    for i in range(z, 0, -1):\n",
    "        if x % i == 0 and y % i == 0:\n",
    "            return i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13495"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# τυχαίες ακέραιες τιμές σε ένα διάστημα\n",
    "import random\n",
    "\n",
    "random.randrange(5000, 20000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.64 ms ± 176 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit \n",
    "# χρονομέτρηση απλοϊκού αλγορίθμου εύρεσης ΜΚΔ\n",
    "\n",
    "a = random.randrange(50000, 200000)\n",
    "b = random.randrange(50000, 200000)\n",
    "x = gcd_naive(a,b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# ταχύτερος αλγόριθμος (αλγόριθμος του Ευκλείδη)\n",
    "def gcd(x, y):\n",
    "    while y:\n",
    "        x, y = y, x % y\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.9 µs ± 105 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "a = random.randrange(50000, 200000)\n",
    "b = random.randrange(50000, 200000)\n",
    "\n",
    "x = gcd(a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Αλγόριθμος του Ευκλείδη για την εύρεση του ΜΚΔ\n",
    "\n",
    "Πολυπλοκότητα: $\\mathcal{O}(\\log{(min(x,y))})$ \n",
    "\n",
    "https://www.baeldung.com/cs/euclid-time-complexity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6 60.0\n"
     ]
    }
   ],
   "source": [
    "# Ελάχιστο κοινό πολλαπλάσιο\n",
    "\n",
    "\n",
    "def lcm(x, y):\n",
    "    return x * y / gcd(x, y)\n",
    "\n",
    "\n",
    "a, b = 30, 12\n",
    "print(gcd(a, b), lcm(a, b))"
   ]
  }
 ],
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