Fraction in lowest terms

Statement

Let $q$ be a rational number. Then there is exactly one pair of integers $a$ and $b$ with $b > 0$, $\gcd(a, b) = 1$ and $q = a / b$.

Remarks

The fraction $a / b$ is the irreducible fraction, or the fraction in lowest terms, of $q$. To find it from any fraction $c / d$ with $d \neq 0$, divide $c$ and $d$ by $\pm \gcd(c, d)$, with the sign that makes the denominator positive. For example, $-6 / -4$ has lowest terms $3 / 2$.

Explanations

L1 human

You can simplify each fraction until the top and the bottom have no common factor larger than $1$. For example, $12 / 18$ becomes $2 / 3$ when you divide both numbers by $6$. The result does not depend on the order of the simplification steps.