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  "body": "\n## Statement\n\nThe [[ring of integers|0JHM6JK7FD]] $\\mathbb{Z}$ is a [[Euclidean domain|0GJ2HRG95T]], with the absolute value $\\delta(n) = |n|$ as Euclidean function.\nThat is, for all $a, b \\in \\mathbb{Z}$ with $b \\neq 0$, there are $q, r \\in \\mathbb{Z}$ with $a = b q + r$ and $0 \\le r \u003c |b|$.\n\n## Remarks\n\nThe proof: let $r$ be the least element of $\\{a - b q : q \\in \\mathbb{Z}\\} \\cap \\mathbb{N}$. Then $r \u003c |b|$, because otherwise $r - |b|$ would be a smaller element of this set.\nThe pair $(q, r)$ with $0 \\le r \u003c |b|$ is unique.\nConsequences: $\\mathbb{Z}$ is a [[principal ideal domain|0RZYDVZ475]], and integers factor uniquely into primes.\n\n## Explanations\n\n### L1 human\n\nThis is division with remainder from school, made precise.\nDividing $17$ by $5$ gives $17 = 3 \\cdot 5 + 2$, and the remainder $2$ is smaller than $5$.\nIt works for negative numbers too: $-17 = (-4) \\cdot 5 + 3$.\n",
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