## How do you find the primitive root of 18?

The order of 1 is 1, the order of 17 is 2, the order of 7 and its inverse 13 is 3, and the order of 5 and its inverse 11 is 6. So the primitive roots mod 18 are 5 and 11.

How do you find the primitive root of 13?

The number of primitive roots mod p is ϕ(p−1). For example, consider the case p = 13 in the table. ϕ(p−1) = ϕ(12) = ϕ(223) = 12(1−1/2)(1−1/3) = 4. If b is a primitive root mod 13, then the complete set of primitive roots is {b1, b5, b7, b11}.

How many primitive roots does Z 19 have?

How many primitive roots does Z<19> have? Explanation: Z<19> has the primitive roots as 2,3,10,13,14 and 15. 13.

### How do you find the primitive root of 17?

Now by problem 7, since Φ(17) = 16, the other primitive roots are the odd powers of 3. In particular one has 3, 33 = 10, 35 = 5, 37 = 11, 39 = 14, 311 = 7, 313 = 12, and 315 = 6 are all primitive roots mod 17.

How do you find the primitive root of 19?

So, if at all 2 has order k modulo 19, and then the possible values of k are 1,2,3,6, and 9. From this, we follow that 18 is the smallest positive integer such that . 2 is a primitive root of 19.

What is the primitive root of 15?

Table of primitive roots

primitive roots modulo order (OEIS: A000010)
14 3, 5 6
15 8
16 8
17 3, 5, 6, 7, 10, 11, 12, 14 16

#### How do you find the primitive root of 25?

Find primitive roots of 4, 25, 18. For 4, the primitive root is 3. For 25, I would first try 2. The powers of 2 are 2, 4, 8, 16, 7, 14, 3, 6, 12, 24 = −1, so 210 ≡ −1 and ord25 2 = 20 = ϕ (25).

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Which among the following values 17/20 38 and 50 does not have primitive roots in the Group G Zn ∗?

Discussion Forum

Que. Which among the following values: 17, 20, 38, and 50, does not have primitive roots in the group G =?
b. 20
c. 38
d. 50

Does 20 have primitive roots?

Since φ(20) = φ(4)φ(5) = 2·4 = 8, it follows immediately that 20 has no primitive root.

## How many primitive roots does Z 19 have Mcq?

How do you find the primitive root of a prime number?

Primitive root of a prime number n modulo n. Given a prime number n, the task is to find its primitive root under modulo n. 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. Return -1 if n is a non-prime number.

When to use primitive roots in proofs?

When primitive roots exist, it is often very convenient to use them in proofs and explicit constructions; for instance, if \\( p \\) is an odd prime and \\( g \\) is a primitive root mod \\( p \\), the quadratic residues mod \\( p \\) are precisely the even powers of the primitive root.

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### What is the primitive root mod of 5?

2 2 is a primitive root mod 5 5, because for every number a a relatively prime to 5, there is an integer z z such that

What are the evenp P powers of primitive roots?

p p are precisely the even powers of the primitive root. Primitive roots are also important in cryptological applications involving the discrete log problem, most notably the Diffie-Hellman key exchange protocol. Z n ∗ = { a ∈ N ⁣: 1 ≤ a < n, gcd ( a, n) = 1 }.