Generating Subgroups in <Z\stackrel{X}{13}> Modulo 13 Under Multiplication

[hitmeoff]

Homework Statement

In the group <Z$$\stackrel{X}{13}$$> of nonzero classes modulo 13 under multiplication, find the subgroup generated by $$\overline{3}$$ and $$\overline{10}$$

Homework Equations

The Attempt at a Solution

Doesnt 3 generate {3,6,9,12} and 10 generate {2,5,10}?

 

[HallsofIvy]

The problem says says "under multiplication" so the subgroup generated by $\overline{3}$ includes all products of $\overline{3}$. You are adding: 3+ 3= 6, etc. 3*3= 9 and 3*9= 27= 1 (mod 13)

 

[Quantumpencil]

3*3 = 9 (13)
3*3*3 =27 = 1 (13)
3*3*3*3 = 81 = 3(13)
3*3*3*3*3 = 243 = 9(13)
3*3*3*3*3*3 = 1 (13)

Do this, and then check that the results are allowed given the constraints you can infer from the order of the Cyclic Group.