Background:
Definition: Let $a_1,a_2,\ldots a_n$ be elements (not all zero) of an integral doain $R.$ A $\textbf{greatest common divisor}$ of $a_1,a_2,\ldots, a_n$ is an element of $d$ of $R$ such that
$(i)$ $d$ divides each other the $a_i;$
$(ii)$ if $c\in R$ and $c$ divides eacch of the $a_i,$ then $c\mid d.$
Theorem 10.13: If $c$ and $d$ are nonzero elements in a unique factorization domain $R,$ then there exist units $u$ and $v$ and irreducibles $p_1,p_2,\ldots, p_k.$ no two of which are associates, such that
$$c=u{p_1}^{m_1}{p_2}^{m_2}\cdots {p_k}^{m_k}\quad\text{and}\quad d=v{p_1}^{n_1}{p_2}^{n_2}\cdots{p_k}^{n_k},$$
where each $m_i$ and $n_j$ is a nonnegative integer. Furthermore,
$c\mid d$ if and only if $m_i\leq n_j$ for each $j=1,2,\ldots,k.$
Theorem 10.18 Let $a_1,a_2,\ldots, a_n$ (not all zero) be elements in a unique factorization domain $R.$ Then $a_1,a_2,\ldots, a_n$ have a greates common divisor in $R.$
Proof: The gcd of any set of elemetns is the gcd of the nonzero memebers of the set, so we may assume that each $a_i$ is nonzero. By Theorem 10.13 there are irreducibles $p_1,\ldots, p_t$ (no two of which are associates), units $u_1,\ldots, u_n,$ and nonnegative integers $m_{ij}$ such that $$a_1=u_1{p_1}^{m_{11}}{p_2}^{m_{12}}{p_1}^{m_{13}}\cdots {p_t}^{m_{1t}}$$ $$a_2=u_2{p_1}^{m_{21}}{p_2}^{m_{22}}{p_1}^{m_{23}}\cdots {p_t}^{m_{2t}}$$ $$\vdots$$ $$a_n=u_n{p_1}^{m_{n1}}{p_2}^{m_{n2}}{p_1}^{m_{n3}}\cdots {p_t}^{m_{nt}}.$$
Let $k_1$ be the smallest exponent that appears on $p_1;$ that is, $k_1$ is the minimum of $m_{11},m_{21},m_{31},\ldots,m_{n1}.$ Similarly, let $k_2$ be the smallest exponent that appears on $p_2$ and so on. Use Theorem 10.13 to verify that $d={p_1}^{k_{1}}{p_2}^{k_{2}}\ldots,{p_t}^{k_{t}}$ is a gcd of $a_1,\ldots,a_n.$
Questions:
In the proof of theorem 10.18 above where at the end it cites theorem 10.13 in that $d={p_1}^{k_{1}}{p_2}^{k_{2}}\ldots,{p_t}^{k_{t}}$ correspond to the $c$ in theorem 10.13, and the $k_i$ correspondsd to the $m_i$ in Theorem 10.13. I can't seem to locate the corresponding $d$ of theorem 10.13 in theorem 10.18. I am not sure if it is each of the $a_i$ or somethng else?
Thank you in advance
