find the order of 3 (mod 14)?

Answer 1

Dude, I answered your question above, and then I noticed all of these other ones. Do your own damn number theory homework.

Answer 2

Order or 3 (mod 14)= LCM( order of 3 (mod 2), order of 3 (mod 7))
3^n=1 (mod 14) ==>>
3*n=1 (mod 2) and 3^n=1 (mod 7)
3^n=1^n=1 (mod 2) order of 3 (mod 2) is 1
3^2=9=2 (mod 7)
3^3=3•2=6=-1 (mod 7)
thus (3^3)^2=(-1)^2=1 (mod 7) thus the order of 3 (mod 7) is 6

Thus the order of 3 (mod 14) is LCM(1,6) = 6

Knowing that 3^6=1 (mod 14) you can check that this is the lowest such n in the following way. If a is the order of 3 (mod 14) then a|6, thus a=1,2,3, or 6 3≠1, 3^2=9≠1, and 3^3=27=-1≠1 (mod 14), thus 6 IS the smallest such n.

Answer 3

6 because 3^6 = 1 (mod 14)