 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ia/paper-4/6e/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The class of

(1, 1)

{(λ, λ) : λ \in (Z / n Z)^{\times}},

φ (n) = n \prod_{p ∣ n} (1 - \frac{1}{p}) .

The unit group acts freely on every pair in

Y

: if

λa \equiv a

and

λb \equiv b

, multiplying the Bezout identity by

λ - 1

gives

λ \equiv 1 (mod n)

φ (n)

. Dividing the answer from part (a) by this size gives

\frac{∣ Y ∣}{φ ( n )} = n p ∣ n \prod \frac{1 - p ^{- 2}}{1 - p ^{- 1}} = n p ∣ n \prod (1 + \frac{1}{p}) .

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