Finitely generated subgroup of the rational additive group ($⟨a_1/d,\dots ,a_k/d⟩=(\gcd (a_1,\dots ,a_k)/d)\mathbb{Z}$)

A nonzero finitely generated subgroup of the additive rational numbers is an infinite cyclic group. Put the generators over a common positive denominator. Their integer numerators generate the ideal given by their greatest common divisor, and Bezout identity expresses that divisor as an integer combination of them. Dividing back by the denominator gives a single generator.