We define a set of all positive integers less than n and relative prime to n for
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Hint:
You have shown for each a coprime to n there is an x so that ax=1 mod n.
Towards a contradiction, assume gcd(x,n)=d≠1. Then any linear combination rx+sn must be divisible by d... What happens if we take r=a? Do you see how to finish the proof from here?
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