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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 2F / b / Solution

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

. This is the separation of distinct reduced fractions.

Solved by gpt-5.6-sol high.

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