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QUESTION
For which of the following moduli does a primitive root exist? In
the cases where one does exist, find one.
(i) 11\hspace{1.5cm} (ii) 30\hspace{1.5cm} (iii) 18\hspace{1.5cm}
(iv) 27\hspace{1.5cm} (v) 4\hspace{1.5cm} (vi) 33.
ANSWER
By th.6.3, a primitive root mod $n$ exists$\Leftrightarrow n$ is a
power of an odd prime, twice a power of an odd prime, 2 or 4. Thus
primitive roots exist for (i), (iii), (iv) and (v), but not for
(ii) or (vi).
For a primitive root mod 11, we need an element of order 10. The
possible orders for $a$ mod 11 are 1,2,5 or 10, so we need to pick
$a\not\equiv\pm1$ mod 11 such that $a^5\not\equiv1$ mod 11. By
trial and error, $a=2$ is a suitable choice.
For a primitive root mod 18, we need, by the argument of th.6.3,
to find an odd element $a$ such that $a$ is a primitive root mod
9. Now $\phi(9)=6$, so we wish to find $a$, odd, such that $o(a)$
mod 9 is not equal to any of 1,2 and 3. By trial and error, 5 is a
suitable choice.
For a primitive root mod 27, th.6.2 shows that we need only pick a
primitive root mod 9. By the above, 5 is a suitable choice. (2
would also do here.)
Finally, for a primitive root mod 4, as $\phi(4)=2$ we need only
pick a residue prime 5to 4 and different from 1. 3 is the only
possible candidate.
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