 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/2f/b/solution

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

Because both fractions are reduced and

q > v

, they are distinct. Their difference is

\frac{p}{q} - \frac{u}{v} = \frac{∣ p v - q u ∣}{q v} .

The numerator is a nonzero integer, so its absolute value is at least one. Therefore

\frac{p}{q} - \frac{u}{v} \geq \frac{1}{q v} > \frac{1}{q ^{2}},

where the strict inequality uses

v < q

 New to topics? Read the docs here!