why is 2 a primitive root of 19 but not of 17? please show an example if at all possible.?

Answer 1

Consider the powers of 2 mod 19:

2^1≡2
2^2≡4
2^3≡8
2^4≡16
2^5≡13
2^6≡7
2^7≡14
2^8≡9
2^9≡18
2^10≡17
2^11≡15
2^12≡11
2^13≡3
2^14≡6
2^15≡12
2^16≡5
2^17≡10
2^18≡1

Note that every single one of the integers coprime to 19 has been generated in this cycle. That's what it means for 2 to be a primitive root mod 19 -- every integer coprime to 19 (which means every integer not divisible by 19, since 19 is prime) is congruent to some power of 2 mod 19. Contrast this with the powers of 2 mod 17:

2^2≡2
2^2≡4
2^3≡8
2^4≡16
2^5≡15
2^6≡13
2^7≡9
2^8≡1
2^9≡2

And it starts repeating. Note that at no point was 3, 5, 6, 7, 10, 11, 12, or 14 generated in this cycle. That's why 2 is not considered a primitive root mod 17 -- there are integers coprime to 7 which cannot be generated by the successive powers of 2.