Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 125 / 4 / c / Solution

For a reduced rational number $x=a/b$ with $b>0$, the height of a rational number isCondition (i) holds because only finitely many coprime integer pairs have bounded maximum.

Condition (iii) also holds. The standard height inequalitygivesCondition (ii) fails: for positive integers $m$,whose absolute value is unbounded. Thus precisely conditions $\boxed{\text{(i) and (iii)}}$ hold. This is consistent with $\mathbb{Q}/n\mathbb{Q}=0$ although the additive group $\mathbb{Q}$ is not finitely generated.