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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 5E / Solution

The Euler theorem states that if

g cd (a, d) = 1

, then

a^{ϕ (d)} \equiv 1 (mod d) .

Let

r_{1}, \dots, r_{ϕ (d)}

be the reduced residue classes modulo

d

. Multiplication by

a

permutes them, so

a^{ϕ (d)} r_{1} \dots r_{ϕ (d)} \equiv r_{1} \dots r_{ϕ (d)} (mod d) .

Their product is a unit and may be cancelled, proving the theorem.

Solved by gpt-5.6-sol high.

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