Hi Sir,

I wanted to make sure what i was doing was right. The question says find the primitive roots of the primes 5 and 7.

For 5

so i obtained 1,2,3,4.

For 7

so obtained 1,2,3,4,5,6.

Is this the correct way to do this? I was getting confused between primitive roots and residue classes. Residue classes are just the numbers co-prime to O(5-1)=O(4), which is 3,1 and for O(7-1)=O(6) is 5,1.

Am i right or on the wrong track?

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Reply: You are getting several notions mixed up. If you fix a modulus , then the

residue classes modulo are represented by all the possible remainders after dividing by , and hence, are

For example, the residue classes modulo 6 are

On the other hand, the residue classes that are *invertible* modulo are represented by the classes where is coprime to . For example, the residue classes that are invertible modulo 6 are

If happens to be a prime, then all non-zero residue classes are invertible modulo

Fermat’s little theorem says that if is a prime, then for any non-zero residue class , we have

So *all* the non-zero residue classes are -th roots of 1 modulo . However, the *primitive* roots of 1 are those for which

but

for any . That is, these are the residue classes that have exact order with respect to multiplication. For example, modulo 5, is a 4-th root of 1, but definitely not a primitive fourth root. Similarly,

but also

So fails as well to be a primitive 4-th root of 1. However, you can check that and *are* primitive 4-th roots of 1.

Now go back to your investigation of the primitive 6-th roots of 1 modulo 7.

By the way, your question indicates that you should read the definitions and examples in the notes more carefully several times.

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