{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nThe number of invertible elements of the ring $\\mathbb{Z}/n\\mathbb{Z}$ of the [[integers modulo n|01N3AYAA6N]] is $\\varphi(n)$, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\n\n## Remarks\n\nBy the [[existence of inverses modulo n|0M67FWZWPQ]], the class $\\overline{a}$ is invertible if and only if $\\gcd(a, n) = 1$. The classes of $1, \\dots, n$ that are coprime to $n$ give $\\varphi(n)$ classes.\nThe invertible elements form a group under multiplication, the group of units $(\\mathbb{Z}/n\\mathbb{Z})^\\times$. It has order $\\varphi(n)$.\nFor example, $(\\mathbb{Z}/8\\mathbb{Z})^\\times = \\{ \\overline{1}, \\overline{3}, \\overline{5}, \\overline{7} \\}$ has $\\varphi(8) = 4$ elements.\n\n## Explanations\n\n### L1 human\n\nModulo $n$, you can divide by exactly $\\varphi(n)$ of the $n$ residues.\nFor $n = 9$, these are the residues of $1, 2, 4, 5, 7, 8$, and $\\varphi(9) = 6$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "11A07"
    ],
    "name": "Number of invertible residues modulo n",
    "role": "proposition",
    "sources": [
      {
        "locator": "Euler's theorem",
        "url": "https://en.wikipedia.org/wiki/Euler%27s_totient_function"
      }
    ],
    "tag": "06RJWZCG95",
    "uses": [
      "01N3AYAA6N",
      "0VV04M41AN",
      "0M67FWZWPQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/06RJWZCG95/",
  "links": [
    {
      "field": "uses",
      "target": "01N3AYAA6N"
    },
    {
      "field": "uses",
      "target": "0VV04M41AN"
    },
    {
      "field": "uses",
      "target": "0M67FWZWPQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/06RJWZCG95.md",
  "msc": [
    "11A07"
  ],
  "name": "Number of invertible residues modulo n",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/06R/06RJWZCG95.md",
  "status": null,
  "tag": "06RJWZCG95",
  "uri": "m2://06RJWZCG95",
  "url": "https://math2.tamnd.com/wiki/number-of-invertible-residues-modulo-n/",
  "writing": {
    "tag": "06RJWZCG95",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}