{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nThen there are positive integers $a$ and $b$ such that $n = a^2 b$ and $b$ is [[square-free|004JW7A2N5]].\n\n## Remarks\n\nThe integers $a$ and $b$ are unique. The number $b$ is the product of the primes $p$ for which the exponent of $p$ in $n$ is odd.\nFor example, $72 = 2^3 \\cdot 3^2 = 6^2 \\cdot 2$, so $a = 6$ and $b = 2$.\nThe proof uses the [[prime factorization|0AEKDFCYFA]] of $n$. Write each exponent $e$ as $e = 2 q + r$ with $r \\in \\{0, 1\\}$.\n\n## Explanations\n\n### L1 human\n\nTake out of $n$ all the pairs of equal primes that you can. The pairs give the square $a^2$.\nAt most one copy of each prime stays, so the rest $b$ is square-free.\nFor example, $50 = 2 \\cdot 5 \\cdot 5 = 5^2 \\cdot 2$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "11A51"
    ],
    "name": "Every integer is a square times a square-free integer",
    "role": "proposition",
    "sources": [
      {
        "locator": "Square-free factors of integers",
        "url": "https://en.wikipedia.org/wiki/Square-free_integer"
      }
    ],
    "tag": "080Q3RPKY6",
    "uses": [
      "004JW7A2N5",
      "0AEKDFCYFA"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/080Q3RPKY6/",
  "links": [
    {
      "field": "uses",
      "target": "004JW7A2N5"
    },
    {
      "field": "uses",
      "target": "0AEKDFCYFA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/080Q3RPKY6.md",
  "msc": [
    "11A51"
  ],
  "name": "Every integer is a square times a square-free integer",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/080/080Q3RPKY6.md",
  "status": null,
  "tag": "080Q3RPKY6",
  "uri": "m2://080Q3RPKY6",
  "url": "https://math2.tamnd.com/wiki/every-integer-is-a-square-times-a-square-free-integer/",
  "writing": {
    "tag": "080Q3RPKY6",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}