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I do not understand the answer starting from "When this is the case, m and n share all but .......", Could anyone explain this for me?
2026-02-25 10:24:19.1772015059
Intersection of sets of multiples of primes
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Note $\ n\in P_j\cap P_k \iff \exists p,q\!:\ pj = n = qk\iff \exists p,q\!:\ \dfrac{j}{k} = \dfrac{q}p,\,$ which is equivalent to $\,j/k\,$ being a quotient of primes $\,q/p\,$ when reduced to lowest terms.
Among $\ \ \dfrac{1}{23},\,\ \ \dfrac{7}{21}\!=\!\dfrac{1}{3},\ \ \dfrac{12}{20}\! =\! \color{#c00}{\dfrac{3}{5}},\,\ \ \dfrac{20}{24}\! =\!\dfrac{5}{6},\,\ \ \dfrac{5}{25}\! =\! \dfrac{1}5\ \ $ only the $\,\rm\color{#c00}{3rd}\,$ reduces to this form.