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Separation of distinct reduced fractions

For distinct reduced fractions

p / q

and

u / v

,

\frac{p}{q} - \frac{u}{v} = \frac{∣ p v - q u ∣}{q v} \geq \frac{1}{q v} .

If

q > v

, this is strictly greater than

q^{- 2}

.

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Solution