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  "body": "\n## Proof\n\nWe divide by the greatest common divisor for existence, and use the generalized Euclid's lemma for uniqueness.\nRecall that two fractions $a / b$ and $c / d$ of [[integers|0C4JS9RF5M]] are the same [[rational number|0NCJ5WXN6Z]] when $a d = b c$.\nLet $q$ be a rational number, and write $q = c / d$ with integers $c$ and $d \\neq 0$.\n\n1. Existence. Let $g = \\gcd(c, d)$ be the [[greatest common divisor|02WF878BH1]]. Since $1$ [[divides|096PW7Q014]] $c$ and $d$, and $c$, $d$ are not both $0$, we get $g \\ge 1$.\n2. Since $g$ divides $c$ and $d$, there are integers $c'$ and $d'$ with $c = g c'$ and $d = g d'$. Since $d \\neq 0$, also $d' \\neq 0$.\n3. We show that $\\gcd(c', d') = 1$. Let $e \\ge 1$ be a common divisor of $c'$ and $d'$, with $c' = e u$ and $d' = e v$.\n4. Then $c = (g e) u$ and $d = (g e) v$, so $g e$ is a common divisor of $c$ and $d$. Hence $g e \\le g$.\n5. Since $g \\ge 1$, step 4 gives $e \\le 1$, so $e = 1$. Every common divisor of $c'$ and $d'$ is therefore at most $1$, and $1$ is one of them. So $\\gcd(c', d') = 1$.\n6. If $d' \u003e 0$, let $a = c'$ and $b = d'$. If $d' \u003c 0$, let $a = -c'$ and $b = -d'$. In both cases $b \u003e 0$.\n7. The integers $a, b$ have the same divisors as $c', d'$, because $k \\mid x$ exactly when $k \\mid -x$. So $\\gcd(a, b) = 1$.\n8. We have $c b = \\pm g c' d' = a d$, with the same sign on both sides. So $a / b = c / d = q$.\n9. Uniqueness. Let $a / b = a' / b' = q$, with $b, b' \u003e 0$ and $\\gcd(a, b) = \\gcd(a', b') = 1$. Then $a b' = a' b$.\n10. So $b$ divides $a b'$. Since $b$ and $a$ are [[coprime|0GFKP8F8AG]], the [[generalized Euclid's lemma|03BAREW1Z5]] gives $b \\mid b'$.\n11. In the same way, $b'$ divides $a' b$ and is coprime to $a'$, so $b' \\mid b$.\n12. Write $b' = b k$ with $k$ an integer. Since $b, b' \u003e 0$, we get $k \u003e 0$, so $k \\ge 1$ and $b' \\ge b$. In the same way, $b \\ge b'$. So $b = b'$.\n13. Then $a b = a' b$ with $b \\neq 0$, so $(a - a') b = 0$ and $a = a'$. So the pair $(a, b)$ is unique.\n\n## Idea\n\nDividing the top and the bottom by their greatest common divisor leaves no common factor, and a sign change makes the bottom positive.\nFor uniqueness, cross-multiplying shows that each denominator divides the other.\nTwo positive numbers that divide each other are equal, and then the numerators agree too.\n",
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