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Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of Fp. (a) For which of the following primes is 2 a primitive root modulo p?a.p = 7b.p = 13c. p = 19d. p = 23

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Answer:

p=13 and p=19.

Step-by-step explanation:

Let <2>={2¹,2²,2³,2⁴,2⁵,2⁶,...} be the set of powers of two modulo p (that is, 2^k=2^kmod p in this set). We can verify if 2 is a primitive root modulo p, by computing (by hand if you like) <2>. If <2> is the set of nonzero elements of Fp, then 2 is a primitive root, if not, then 2 is not a primitive root.

In F_7, <2>={2,4,1}≠F_7\{0}

In F_13, <2>={2,4,8,3,6,12,11,9,2,10,7,1}=F_13\{0}

In F_19, <2>={2,4,8,16,13,7,14,9,18,17,15,11,3,6,12,5,10,1}=F_19\{0}

In F_23, <2>={2,4,8,16,9,18,13,3,6,12,1}≠F_23\{0}

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