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# Complete residue system? watch

1. There's a question I've been trying to tackle in my tutorial sheet that's really bugging me, that I can't understand even with the provided solutions.

Here's the question with the solution below it.

I lose track when he changes to mod(p-1). I assume the solutions are using some kind of lemma or theorem taught in the lectures, but I missed a few days and no bells are ringing.

Thanks for any help.

Edit: Just noticed the title doesn't really fit my question, I do know what a complete residue system is.
2. (Original post by jamie092)
There's a question I've been trying to tackle in my tutorial sheet that's really bugging me, that I can't understand even with the provided solutions.

Here's the question with the solution below it.

I lose track when he changes to mod(p-1). I assume the solutions are using some kind of lemma or theorem taught in the lectures, but I missed a few days and no bells are ringing.

Thanks for any help.

Edit: Just noticed the title doesn't really fit my question, I do know what a complete residue system is.
Raising the primitive root to a power is cyclic with the period being of length p-1. For example, g^0=g^(p-1)=g^(2(p-1))=1 (mod p). Furthermore, only natural indices congruent to 0 (mod p-1) will have this residue mod p. Simiilar statements apply for all other residues mod p such that the residue is coprime to p.

Essentially we have a bijection from the residues mod p-1 to the residues of p which are coprime to p.
3. (Original post by Magu1re)
Raising the primitive root to a power is cyclic with the period being of length p-1. For example, g^0=g^(p-1)=g^(2(p-1))=1 (mod p). Furthermore, only natural indices congruent to 0 (mod p-1) will have this residue mod p. Simiilar statements apply for all other residues mod p such that the residue is coprime to p.

Essentially we have a bijection from the residues mod p-1 to the residues of p which are coprime to p.
Ah ok, thanks a lot!

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