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   "cell_type": "markdown",
   "id": "6cfdb9dc-df35-4ac5-9fa9-da7cae02d2b8",
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   "source": [
    "### Euler's product formula\n",
    "\n",
    "$\\phi(n)=n\\prod_{p|n}\\left(1-\\frac{1}{p}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "84b0bbc4-48f4-4dc9-a1a9-0cab8f68829a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n",
      "12\n"
     ]
    }
   ],
   "source": [
    "print(euler_phi(12))\n",
    "print(euler_phi(13))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0be1d436-5075-4f2f-96ec-e225179209a7",
   "metadata": {},
   "source": [
    "### Euler's theorem\n",
    "\n",
    "$gcd(a, n)=1$\n",
    "\n",
    "$a^{\\phi(n)}\\equiv1 \\bmod n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "34b39cb0-1bc0-4f35-8c9f-ff47d575bc9e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "print(gcd(7,13))\n",
    "print(pow(7,euler_phi(13),13))"
   ]
  }
 ],
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