Given a rectangle of size n x m, find the minimum number of integer-sided squares that tile the rectangle.
 
Example 1:

Input: n = 2, m = 3
Output: 3
Explanation: 3 squares are necessary to cover the rectangle.
2 (squares of 1x1)
1 (square of 2x2)
Example 2:

Input: n = 5, m = 8
Output: 5

Example 3:

Input: n = 11, m = 13
Output: 6

 
Constraints:

1 <= n <= 13
1 <= m <= 13

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:    \n    # def upperBound(self, n, m):\n    #     if n == 0 or m == 0:\n    #         return 0\n    #     elif m > n:\n    #         return self.upperBound(m, n)\n    #     # so that m < n\n    #     else:\n    #         return n // m + self.upperBound(m, n % m)\n    \n    def findSum(self, n, m, k):\n        res = []\n        s = m*n\n        hi = min(n, m)\n        border = max(n, m)\n        \n        path = [1] * k\n        while path[0] <= hi:\n            # find the next available sum\n            # numbers are monotone non-decreasing in path\n            i = k-1\n            while i >= 0 and path[i] >= hi:\n                i -= 1\n            if i == -1 or path[i] >= hi:\n                return res\n            \n            path[i] += 1\n            path[i+1:] = [path[i]] * (k-i-1)\n            if path[k-1] + path[k-2] > border:\n                path[i:] = [hi] * (k-i)\n                continue\n            \n            if sum(x*x for x in path) == s:\n                x = path[:]\n                x.reverse()\n                res.append(x)\n        \n        return res\n    \n    def hasNoOverLap(self, x1, y1, s1, x2, y2, s2):\n        if x1+s1-1 < x2 or y1+s1-1 < y2 or x2+s2-1 < x1 or y2+s2-1 < y1:\n            return True\n        else:\n            return False\n        \n    def nextPos(self, placement, n, m, size):\n        if not placement:\n            return (0,0)\n        \n        for i in range(n-size+1):\n            for j in range(m-size+1):\n                if all(self.hasNoOverLap(i, j, size, x, y, z) for x, y, z in placement):\n                    return i, j\n        \n        return -1, -1\n            \n        \n    \n    def canPlace(self, sizes, n, m, placement, memo):\n        if len(sizes) == 0:\n            return True\n        h = tuple(placement)\n        if h in memo:\n            return memo[h]\n        \n        for k, s in enumerate(sizes):\n            i, j = self.nextPos(placement, n, m, s)\n            if i == -1:\n                continue\n            placement.append((i,j, s))\n            if self.canPlace(sizes[:k] + sizes[k+1:], n, m, placement, memo):\n                memo[h] = True\n                return True\n            placement.pop()\n        \n        memo[h] = False\n        return False\n        \n        \n        \n        \n    def tilingRectangle(self, n: int, m: int) -> int:\n        if n % m == 0:\n            return n // m\n        if m % n == 0:\n            return m // n\n    \n        for i in range(2, 10):\n            res = self.findSum(n, m, i)\n            if any(self.canPlace(sizes, n, m, [], {}) for sizes in res):\n                return i\n        \n        \n        \n        \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'tilingRectangle'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 11104