{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $n = \\sum_{i=0}^{k} d_i 10^i$ with decimal digits $d_0, \\dots, d_k$. Let $S = d_0 + d_1 + \\dots + d_k$ be the digit sum.\n2. Since $10 - 1 = 9 = 9 \\cdot 1$, the number $9$ [[divides|096PW7Q014]] $10 - 1$. So $10 \\equiv 1 \\pmod 9$ by the definition of [[congruence|0TPW25CBN6]].\n3. Let $0 \\le i \\le k$. By step 2 and [[compatibility with powers|0J5VGJYYC3]], $10^i \\equiv 1^i = 1 \\pmod 9$.\n4. Since $d_i \\equiv d_i \\pmod 9$, step 3 and [[compatibility with multiplication|072D5G4SK6]] give $d_i 10^i \\equiv d_i \\pmod 9$.\n5. Apply [[compatibility with addition|00FWBN7F0Y]] $k$ times to the $k + 1$ congruences of step 4. This gives $n \\equiv S \\pmod 9$.\n6. So there is an integer $t$ with $n - S = 9 t$.\n7. If $9 \\mid S$, say $S = 9 s$, then $n = S + 9 t = 9 (s + t)$, so $9 \\mid n$.\n8. If $9 \\mid n$, say $n = 9 m$, then $S = n - 9 t = 9 (m - t)$, so $9 \\mid S$.\n9. By steps 7 and 8, $9 \\mid n$ if and only if $9 \\mid d_0 + d_1 + \\dots + d_k$.\n\n## Idea\n\nEach power of $10$ leaves remainder $1$ on division by $9$, because $10 = 9 + 1$. So each digit counts for itself modulo $9$, and the number and its digit sum differ by a multiple of $9$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "methods": [
      "0NTF6DMVAP"
    ],
    "msc": [
      "11A63"
    ],
    "name": "Proof of the divisibility rule for 9",
    "proves": "0ESHD7ZQDX",
    "sources": [
      {
        "locator": "Case where all digits are summed",
        "url": "https://en.wikipedia.org/wiki/Divisibility_rule"
      }
    ],
    "tag": "SJWH2V43J2",
    "uses": [
      "0TPW25CBN6",
      "096PW7Q014",
      "0J5VGJYYC3",
      "072D5G4SK6",
      "00FWBN7F0Y"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/SJWH2V43J2/",
  "links": [
    {
      "field": "proves",
      "target": "0ESHD7ZQDX"
    },
    {
      "field": "uses",
      "target": "0TPW25CBN6"
    },
    {
      "field": "uses",
      "target": "096PW7Q014"
    },
    {
      "field": "uses",
      "target": "0J5VGJYYC3"
    },
    {
      "field": "uses",
      "target": "072D5G4SK6"
    },
    {
      "field": "uses",
      "target": "00FWBN7F0Y"
    },
    {
      "field": "methods",
      "target": "0NTF6DMVAP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SJWH2V43J2.md",
  "msc": [
    "11A63"
  ],
  "name": "Proof of the divisibility rule for 9",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SJW/SJWH2V43J2.md",
  "status": null,
  "tag": "SJWH2V43J2",
  "uri": "m2://SJWH2V43J2",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-divisibility-rule-for-9/",
  "writing": {
    "tag": "SJWH2V43J2",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}