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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-23/5/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 5 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

a / q

with

1 \leq q \leq Q

0 \leq a < q

and

(a, q) = 1

, regarded in

R / Z

. The fraction zero appears as

0/1

; one is the same circle point and is not added a second time. For two distinct such points, the ordinary difference and its possible wrapped complement are nonzero integer multiples of

1/ (q q^{'})

. Hence

∥ a / q - a^{'} / q^{'} ∥_{R / Z} \geq 1/ (q q^{'}) \geq Q^{- 2} .

(1)

This proves the required separation of the Farey fractions.

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