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   "source": [
    "# [Digit Cancelling Fractions](https://projecteuler.net/problem=33)\n",
    "\n",
    "The search term for this concept is an [anomalous cancellation](https://en.wikipedia.org/wiki/Anomalous_cancellation). We can write a function that checks if an anomalous cancellation can happen by storing the digits of the numerator and denominator into four variables total, then checking four separate cases for if a digit can be \"cancelled.\""
   ]
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   "id": "d4e94963",
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   "source": [
    "def digits(n):\n",
    "    return (n // 10, n % 10)\n",
    "\n",
    "\n",
    "def can_cancel_digits(n, d):\n",
    "    x, y = digits(n)\n",
    "    z, w = digits(d)\n",
    "\n",
    "    f = QQ(n/d)\n",
    "\n",
    "    if x == z and w != 0 and y/w == f:\n",
    "        return True\n",
    "    elif x == w and y/z == f:\n",
    "        return True\n",
    "    elif y == z and w != 0 and x/w == f:\n",
    "        return True\n",
    "    elif y == w and w != 0 and x/z == f:\n",
    "        return True\n",
    "\n",
    "    return False"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a68309e9",
   "metadata": {},
   "source": [
    "We're only dealing with two-digit numerators and denominators, so this is easy to brute force."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "756614f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(16, 64), (19, 95), (26, 65), (49, 98)}"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fractions = set()\n",
    "for n in range(10, 100):\n",
    "    for d in range(n + 1, 100):\n",
    "        if can_cancel_digits(n, d):\n",
    "            fractions.add((n, d))\n",
    "\n",
    "fractions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d3fb43b",
   "metadata": {},
   "source": [
    "There's only four such fractions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "90f47a7f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prod(QQ(n/d) for (n, d) in fractions).denominator()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1514f872",
   "metadata": {},
   "source": [
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
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