Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 3 / 1E / Solution

$A={\mathbb{Z}}^2/⟨(6,9)⟩$. The class of $(2,3)$ has order $3$, so $A$ has torsion and is not free. A unimodular change of basis (Smith normal form, using $\gcd (6,9)=3$) gives $A\cong \mathbb{Z}\oplus \mathbb{Z}/3\mathbb{Z}$.

Solved by gpt-5.6-sol high.