{
  "aliases": [
    "bezout-lemma",
    "bezouts-lemma"
  ],
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  "body": "\n## Statement\n\nLet $a$ and $b$ be integers, and let $d = \\gcd(a, b)$ be their [[greatest common divisor|02WF878BH1]].\nThen there are integers $x$ and $y$ such that $a x + b y = d$.\n\n## Remarks\n\nThe integers $x$ and $y$ are not unique. If $(x, y)$ is a solution, then $(x + k b / d, \\; y - k a / d)$ is also a solution for each integer $k$, when $d \\neq 0$.\nThe set of all integers $a x + b y$ is the set of the multiples of $d$.\nThe [[extended Euclidean algorithm|0964YQFSA8]] computes one pair $(x, y)$.\nFor example, $\\gcd(12, 42) = 6$ and $12 \\cdot (-3) + 42 \\cdot 1 = 6$.\n\n## Explanations\n\n### L1 human\n\nWith coins of value $a$ and $b$, and if you can give and take back coins, you can pay exactly the amount $\\gcd(a, b)$.\nFor example, with coins of $4$ and $6$, you pay $2 = 2 \\cdot 4 - 1 \\cdot 6$.\nYou cannot pay $1$ this way, because each sum $4 x + 6 y$ is even.\n",
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    "aliases": [
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    "kind": "statement",
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    "name": "Bézout's identity",
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      "wikidata": "Q513028",
      "wikipedia": "Bézout's identity"
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    "sources": [
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        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity"
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    "tag": "0DQVHMYAGA",
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  "markdown": "https://math2.tamnd.com/node/0DQVHMYAGA.md",
  "msc": [
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  "name": "Bézout's identity",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DQ/0DQVHMYAGA.md",
  "status": null,
  "tag": "0DQVHMYAGA",
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    "tag": "0DQVHMYAGA",
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