Question

(3) (6') Prove that 3 is a primitive root of 7 .

Answer 1

To prove that 3 is a primitive root of 7, we need to show that the powers of 3, modulo 7, cycles through all the possible remainders except for 0.

Step-by-step explanation:

To prove that 3 is a primitive root of 7, we need to show that the powers of 3, modulo 7, cycles through all the possible remainders except for 0. Let's calculate the powers of 3 modulo 7:

1. 3¹ mod 7 = 3
2. 3² mod 7 = 2
3. 3³ mod 7 = 6
4. 3⁴ mod 7 = 4
5. 3⁵ mod 7 = 5
6. 3⁶ mod 7 = 1
7. 3⁷ mod 7 = 3
8. 3⁸ mod 7 = 2
9. ...

We observe that the remainders repeat after every 6 powers, covering all the possible remainders except for 0. Therefore, 3 is a primitive root of 7.