Chapter 6.2, Problem 9E

For any $a\in \mathbb{Z},$ let ${\left[a\right]}_6$ denote $\left[a\right]$ in ${\mathbb{Z}}_6$ and let ${\left[a\right]}_{2_{}}$ denote $\left[a\right]$ in ${\mathbb{Z}}_2$.

a. Prove that the mapping $\theta :{\mathbb{Z}}_6\to {\mathbb{Z}}_2$ defined by $\theta \left({\left[a\right]}_6\right)={\left[a\right]}_2$ is a homomorphism.

b. Find ker $\theta$.