using mods what is the remainder of 14^485 when divided by 3?

Answer 1

14^485 mod 3

Well 14 mod 3 ≡ 2, or -1 if you prefer
14^2 ≡ 1 mod 3
Thus (14^2)^k ≡ 1 mod 3 for any natural k

So 14^485 = 14*(14^2)^242
14*(14^2)^242 ≡ (2)*(1)^242 mod 3
= 2*1 mod 3
= 2 mod 3

Answer 2

Answer: 2

14=2 mod 3
14^2=1 mod 3
14^484=1 mod3
14^485=1*2=2 mod3

Answer 3

14 mod3 ≡ -1 mod3
14^485 mod3 ≡ -1 mod3 ≡ 2 mod3, since (-1)^485 = -1
Therefore, the remaider is 2.