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  "body": "\n## Statement\n\nLet $H$ be a [[subgroup|0X1JXT47GA]] of the [[additive group of integers|0BSCJA5G53]] $(\\mathbb{Z}, +)$.\nThen there is exactly one integer $n \\geq 0$ with $H = n\\mathbb{Z} = \\{nk : k \\in \\mathbb{Z}\\}$.\n\n## Remarks\n\nWhen $H \\neq \\{0\\}$, the integer $n$ is the smallest positive element of $H$.\nFor $m, n \\geq 0$, $m\\mathbb{Z} \\subseteq n\\mathbb{Z}$ holds exactly when $n$ divides $m$.\nA consequence is that, for integers $a$ and $b$, the subgroup $a\\mathbb{Z} + b\\mathbb{Z}$ is $d\\mathbb{Z}$ with $d = \\gcd(a, b)$, which gives Bezout's identity.\n\n## Explanations\n\n### L1 human\n\nEach subgroup of the integers is the set of multiples of one number.\nFor example, the set of all sums $4x + 6y$ with integers $x, y$ is the set of multiples of $2$.\nThe number $0$ gives the trivial subgroup $\\{0\\}$.\n",
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