(a) Consider the homomorphism $f: \mathbb{Z}^{3} \rightarrow \mathbb{Z}^{4}$ given by

$$f(a, b, c)=(a+2 b+8 c, 2 a-2 b+4 c,-2 b+12 c, 2 a-4 b+4 c)$$

Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of $\mathbb{Z}^{4}$ by the image of this homomorphism as an abstract abelian group.

(b) Give the definition of a Euclidean domain.

Fix a prime $p$ and consider the subring $R$ of the rational numbers $\mathbb{Q}$ defined by

$$R=\{q / r \mid \operatorname{gcd}(p, r)=1\}$$

where 'gcd' stands for the greatest common divisor. Show that $R$ is a Euclidean domain.