Question

Is 3 a primitive root of 7? Select one:
a. Cant say
b. True
c. False
d. May be?

Answer 1

Yes

Step-by-step explanation:

3 is the primitive root of 7 because:

$3^1 = 3 = 3^0\times 3$ ≡ 1 × 3 = 3 ≡ 3 (mod 7)

$3^2 = 9 = 3^1\times 3$ ≡ 3 × 3 = 9 ≡ 2 (mod 7)

$3^3 = 27 = 3^2\times 3$ ≡ 2 × 3 = 6≡ 6 (mod 7)

$3^4 = 81 = 3^3\times 3$ ≡ 6 × 3 = 18≡ 4 (mod 7)

$3^5 = 243 = 3^4\times 3$ ≡ 4 × 3 = 12≡ 5 (mod 7)

$3^6 = 729 = 3^5\times 3$ ≡ 5 × 3 = 15≡ 1 (mod 7)

$3^7 = 2187 = 3^6\times 3$ ≡ 1 × 3 = 3≡ 3 (mod 7)

Since,we have achieved all values from 1 to 6 in our residue results.