{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# L'arithmétique"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quotient et reste\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n",
      "7\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "dividende = 47\n",
    "diviseur = 8\n",
    "quotient = dividende // diviseur\n",
    "reste = dividende % diviseur\n",
    "print(quotient)\n",
    "print(reste)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Liste des diviseurs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def diviseurs(nombre):\n",
    "    \"retourne la liste des diviseurs.\"\n",
    "    limite=int(math.sqrt(nombre)+1)\n",
    "    L_couples = [(x,nombre//x) for x in range(1,limite) if nombre%x==0]\n",
    "    #print(L_couples)\n",
    "    ensemble_diviseurs=set() # ensemble des diviseurs\n",
    "    for (a,b) in L_couples :\n",
    "        ensemble_diviseurs.add(a)\n",
    "        ensemble_diviseurs.add(b)\n",
    "    L_diviseurs=sorted(ensemble_diviseurs) #conversion en liste ordonnées.\n",
    "    return L_diviseurs\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3, 6, 9, 18]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diviseurs(18)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tester si un nombre est premier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def est_premier(nombre) :\n",
    "    \"Un nombre est premier s'il a deux diviseurs\"\n",
    "    if len(diviseurs(nombre))==2 :\n",
    "        return True\n",
    "    else :\n",
    "        return False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "est_premier(17)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Décomposition en facteurs premiers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def decomposition(nombre):\n",
    "    \"retourne la liste des facteurs premiers\"\n",
    "    L=diviseurs(nombre)\n",
    "    if len(L)==1 :\n",
    "        return [1]\n",
    "    Decomposition=[]\n",
    "    while len(L)>1 :\n",
    "        L.pop(0)\n",
    "        p=L[0]\n",
    "        Decomposition.append(p)\n",
    "        L=[k//p for k in L if k%p==0]\n",
    "    return Decomposition\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 3, 3607, 3803]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "decomposition(123456789)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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