ϕ (p − 1) = ϕ (12) = ϕ (2 2 3) = 12(1 − 1/2)(1 − 1/3) = 4. Given that 2 is a primitive root of 59, find 17 other primitive roots of 59. 2) Next, we find a primitive root mod 82. Use the fact: If g is a primitive root mod p, then one of g and g+p (whichever one is odd) is a primitive root mod 2p. Then, 6 + 41 = 47 is a primitive root mod 82 (since 47 is odd). Thus, first find a small primitive root, i.e., find an a such that the smallest integer k that satisfies a k mod 13 = 1 is k = m – 1 = 12. For such a prime modulus generator all primitive roots produce full cycles. To ﬁnd the other prinitive roots, use the table that was written down today in class.] If b is a primitive root mod 13, th en the complete set of primitive roots is {b 1, b 5, b 7, b 11}. (1) Find the index of 5 relative to each of the primitive roots of 13. It is easily verified that 2 k mod 13 = 2, 4, 8, 3, 6, 12, 11, 9, … We see from the table that 2 is a primitive root mod 13… Since primitive roots are of the form 3 i where gcd(i, φ (17)) = 1, the primitive roots are 3, 10, 11, 14, 7, 12, 6, 5 (d) We showed above that the primitive roots of 18 are 5 and 11. The number of primitive roots mod p is ϕ (p − 1). But my question is how can I use this information to deduce that the product of all the primitive roots mod 13 is congruent to 1 mod 13. Suppose that we have a primitive root, g. For example, 2 is a primitive root of 59. In fact, I have shown that g^11 is a primitive root mod 13. I'm aware of the condition for k to such that g^k is a primitive root mod 13. (3) Find the number of roots mod 13 for each of the following polynomials: (a) x 2 + 3 x + 2 (b) x … For example, consider the case p = 13 in the table. By (1), 6 is a primitive root mod 41. We know that 3, 5, 7, 11, 13… Then it turns out for any integer relatively prime to 59-1, let's call it b, then \$2^b (mod 59)\$ is also a primitive root of 59. Example 1. Primitive root of a prime number n is an integer r between[1, n-1] such that the values of r^x(mod n) where x is in range[0, n-2] are different. [Hint: Recall that 2 is a primitive root modulo 13.