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  "body": "\n## Statement\n\nLet $k$ be a natural number.\nThe [[divisor function|0YEW47VJ0J]] $\\sigma_k$ is a [[multiplicative function|0AP80TT660]].\nThat is, $\\sigma_k(mn) = \\sigma_k(m) \\sigma_k(n)$ for all coprime positive integers $m$ and $n$.\n\n## Remarks\n\nThe proof: for coprime $m$ and $n$, each divisor of $mn$ is a unique product $d e$ with $d \\mid m$ and $e \\mid n$.\nThe function $\\sigma_k$ is not completely multiplicative: $\\sigma_0(4) = 3$, but $\\sigma_0(2)^2 = 4$.\nAt a prime power, $\\sigma_k(p^e) = 1 + p^k + p^{2k} + \\dots + p^{ek}$. Multiplicativity then gives $\\sigma_k(n)$ for all $n$.\n\n## Explanations\n\n### L1 human\n\nFor example, $\\sigma(4) = 7$ and $\\sigma(3) = 4$, so $\\sigma(12) = 28$.\nThis works because $4$ and $3$ share no prime factor. Each divisor of $12$ is a divisor of $4$ times a divisor of $3$.\n",
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