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  "body": "\n## Statement\n\nLet $n$ be a [[composite number|0N3ADXHNTQ]].\nThen $n$ has a prime factor $p$ with $p^2 \\le n$, that is, $p \\le \\sqrt{n}$.\n\n## Remarks\n\nThe [[smallest prime factor|0S2SYHTWPY]] $p$ of $n$ has this property. Write $n = p m$. Then $m \u003e 1$, and each prime factor of $m$ is at least $p$, so $m \\ge p$ and $n \\ge p^2$.\nThis result justifies [[trial division|0Z962ECCXR]]: to test whether $n \u003e 1$ is prime, it is enough to test the primes $p \\le \\sqrt{n}$.\nFor example, $\\sqrt{97} \u003c 10$ and none of $2, 3, 5, 7$ divides $97$, so $97$ is prime.\n\n## Explanations\n\n### L1 human\n\nIf $n = a b$ with both factors larger than $\\sqrt{n}$, then $a b \u003e n$, which is impossible.\nSo one of the two factors is at most $\\sqrt{n}$, and a prime factor of that factor is also at most $\\sqrt{n}$.\n",
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