How to show this is a homomorphism?

[SMA_01]

How to show this is a homomorphism?

Homework Statement

Let θ: Z6→Z2 be given by θ(x)=the remainder of x when divided by 2 (as in the division algorithm)

Homework Equations

The Attempt at a Solution

I am stuck, this is all I have:

Let m,n be in Z6
θ(m +6 n)...

I'm not sure how to proceed. Any help is appreciated. Thanks

 

[kru_]

If m is an element of Z6. What does it look like?

 

[SMA_01]

@kru- What do you mean? I imagine Z6 to look like a circle that starts and ends at 6...not sure if this is very accurate though.

 

[Leb]

I think kru_ is asking you how would you express an element of 6Z or 2Z or (any integer)Z in the general form (hint, odd integers are expressed as 2k + 1, where k is in Z).

From there, you should think what is the remainder of any element in 6Z divided by 2 and simply use the definition of homomorphism of groups.

If it is Z₆ and Z₂ (or Z(mod6) and Z(mod2) in other words) you are talking about, then you follow a very similar logic, just with a little more writing required in your proof.