Let $r \in \mathbb{Z}_{\ge 0}$. Let $p_1,\dots,p_r$ be pairwise distinct prime numbers, and let $\alpha_i,\beta_i \in \mathbb{Z}_{\ge 0}$ for each $i \in \{1,\dots,r\}$. Suppose $a,b \in \mathbb{N}$ satisfy
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\begin{align*}
a = \prod_{i=1}^r p_i^{\alpha_i}
\end{align*}
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and
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\begin{align*}
b = \prod_{i=1}^r p_i^{\beta_i}.
\end{align*}
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When $r=0$, the products are understood to be the empty product $1$. Then