{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0QTHB6PE52"
    }
  ],
  "body": "\n## Statement\n\nLet $n \\geq 1$ be an integer.\nThen the [[order|0B85S2V2DR]] of the group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is $\\varphi(n)$.\nHere $\\varphi(n)$ is the number of integers $a$ with $0 \\leq a \\leq n - 1$ and $\\gcd(a, n) = 1$.\n\n## Remarks\n\nThe proof is a count, because the [[units of Z/nZ are the residues coprime to n|0GXNHEHQNA]].\nFor a prime $p$, the order is $\\varphi(p) = p - 1$.\nFor example, $(\\mathbb{Z}/9\\mathbb{Z})^\\times = \\{1, 2, 4, 5, 7, 8\\}$ has order $\\varphi(9) = 6$.\nWith [[Lagrange's theorem|3C53TD]], this gives Euler's theorem: $a^{\\varphi(n)} \\equiv 1 \\pmod{n}$ for each $a$ coprime to $n$.\n\n## Explanations\n\n### L1 human\n\nThe size of the group of units modulo $n$ is the number of residues coprime to $n$.\nEuler's totient function counts exactly these residues.\nSo the group has $\\varphi(n)$ elements.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "11A07",
      "11A25"
    ],
    "name": "Order of the group of units of Z/nZ is Euler's totient",
    "role": "proposition",
    "sources": [
      {
        "locator": "Structure",
        "url": "https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n"
      },
      {
        "locator": "n = 9",
        "url": "https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n"
      }
    ],
    "tag": "0S7495H4HM",
    "uses": [
      "0GXNHEHQNA",
      "0SEX9F9J5Z"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0S7495H4HM/",
  "links": [
    {
      "field": "uses",
      "target": "0GXNHEHQNA"
    },
    {
      "field": "uses",
      "target": "0SEX9F9J5Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0S7495H4HM.md",
  "msc": [
    "11A07",
    "11A25"
  ],
  "name": "Order of the group of units of Z/nZ is Euler's totient",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0S7/0S7495H4HM.md",
  "status": null,
  "tag": "0S7495H4HM",
  "uri": "m2://0S7495H4HM",
  "url": "https://math2.tamnd.com/wiki/order-of-the-group-of-units-of-z-nz-is-eulers-totient/",
  "writing": {
    "tag": "0S7495H4HM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}