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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 5D / d / i / Solution

Write

d = q r + s

with

0 \leq s < r

. Since

x^{d} = e \in H

and

x^{r} \in H

,

x^{s} = x^{d} (x^{r})^{- q} \in H .

The minimality of the positive integer

r

forces

s = 0

. Therefore

r ∣ d .

Solved by gpt-5.6-sol high.

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