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Show that the numbers 1, 3, 3^2 , . . . , 3^15 and 0 for...

Show that the numbers 1, 3, 3^2 , . . . , 3^15 and 0 for a complete system of residues (mod 17). Do the numbers 1, 2, 2^2 , . . . , 2^15 and 0 constitute a complete system of residues (mod 17)?

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Answer #1

We can stop at this point. Since and is the order of the group of invertible elements under multiplication modulo 17, the order of must be a divisor of so it must be one of

As it is not it must be

Thus, must all be unique elements under multiplication modulo 17

Thus, forms a complete system of residues modulo 17

On the other hand, we can check that so this is not a complete system of residues

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