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  "body": "\n## Proof\n\nWe use Bézout's identity.\n\n1. Let $n$, $a$ and $b$ be integers with $n \\mid ab$, and let $n$ and $a$ be [[coprime|0GFKP8F8AG]], so $\\gcd(n, a) = 1$.\n2. By [[Bézout's identity|0DQVHMYAGA]], there are integers $x$ and $y$ with $n x + a y = 1$.\n3. Multiplying by $b$ gives $b = n b x + a b y$.\n4. Since $n$ [[divides|096PW7Q014]] $ab$, there is an integer $k$ with $ab = n k$. Then $a b y = n k y$.\n5. So $b = n b x + n k y = n (b x + k y)$, and $b x + k y$ is an integer.\n6. Hence $n \\mid b$.\n\n## Idea\n\nCoprimality gives $nx + ay = 1$. Multiplying by $b$ writes $b$ as a sum of two multiples of $n$: one is visibly a multiple, and the other is a multiple of $ab$.\n",
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