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  "body": "\n## Summary\n\nThe extended Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] $d$ of two integers $a$ and $b$, and also integers $x$ and $y$ such that $a x + b y = d$.\nIt does the steps of the [[Euclidean algorithm|0TRNT4X7J0]].\nAt each step, it writes the new remainder as an integer combination of $a$ and $b$.\n\n## Remarks\n\nThe coefficients $x$ and $y$ prove [[Bézout's identity|0DQVHMYAGA]] for $a$ and $b$.\nWhen $\\gcd(a, n) = 1$, the coefficient $x$ of $a$ is a [[modular multiplicative inverse|00GXTQMY8Z]] of $a$ modulo $n$.\n\n## Explanations\n\n### L1 human\n\nFor $a = 48$ and $b = 18$, the Euclidean algorithm gives $12 = 48 - 2 \\cdot 18$ and $6 = 18 - 1 \\cdot 12$.\nPut the first equation into the second: $6 = 18 - (48 - 2 \\cdot 18) = 3 \\cdot 18 - 1 \\cdot 48$.\nSo $x = -1$ and $y = 3$ give $48 x + 18 y = 6$.\n",
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