Final answer:
The statement that ` 10 is a primitive root of 17` is FALSE because the discrete logarithm of 12 base 10 with modulus 17 is 15.
The answer is option ⇒b. False.
Step-by-step explanation:
To show that 10 is a primitive root of 17, we need to check whether the powers of 10, when taken modulo 17, produce all the non-zero residues (1, 2, 3, ..., 16).
Let's calculate the powers of 10 modulo 17:
10¹ ≡ 10 (mod 17)
10² ≡ 100 ≡ 16 (mod 17)
10³ ≡ 1000 ≡ 13 (mod 17)
10⁴ ≡ 10000 ≡ 4 (mod 17)
10⁵ ≡ 100000 ≡ 7 (mod 17)
10⁶≡ 1000000 ≡ 11 (mod 17)
10⁷ ≡ 10000000 ≡ 9 (mod 17)
10⁸ ≡ 100000000 ≡ 6 (mod 17)
10⁹ ≡ 1000000000 ≡ 14 (mod 17)
10¹⁰ ≡ 10000000000 ≡ 3 (mod 17)
10¹¹ ≡ 100000000000 ≡ 8 (mod 17)
10¹² ≡ 1000000000000 ≡ 5 (mod 17)
10¹³ ≡ 10000000000000 ≡ 15 (mod 17)
10¹⁴ ≡ 100000000000000 ≡ 2 (mod 17)
10¹⁵ ≡ 1000000000000000 ≡ 12 (mod 17)
10¹⁶ ≡ 10000000000000000 ≡ 1 (mod 17)
As we can see, the powers of 10 modulo 17 produce all the non-zero residues (1, 2, 3, ..., 16). This means that 10 is a primitive root of 17.
To calculate the discrete logarithm of 12 base 10 (with prime modulus 17), we need to find the exponent k such that 10ᵏ ≡ 12 (mod 17).
By checking the powers of 10 modulo 17 above, we can see that 10¹⁵ ≡ 12 (mod 17). Therefore, the discrete logarithm of 12 base 10 with modulus 17 is 15.
Based on the above calculations, the answer is:
b. False