 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ib/paper-3/1e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

A = 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

g cd (6, 9) = 3

) gives

A ≅ Z \oplus Z /3 Z

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