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Question Number 73553 by 01 last updated on 13/Nov/19
give: a,b,c>0  if a∣b and b∣c  prove : a∣c
$$\mathrm{give}:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{if}\:\mathrm{a}\mid\mathrm{b}\:\mathrm{and}\:\mathrm{b}\mid\mathrm{c} \\ $$$$\mathrm{prove}\::\:\mathrm{a}\mid\mathrm{c} \\ $$
Answered by Rio Michael last updated on 13/Nov/19
a∣b ⇔ ∃ k∈Z :  b = ka ......(i)  b∣ c ⇔ ∃ m ∈Z :  c = bm.....(ii)  eqn (i) in (ii) ⇒    c = (ka)m    c = km a    if k,m∈ Z the km∈Z  hence a∣c
$${a}\mid{b}\:\Leftrightarrow\:\exists\:{k}\in\mathbb{Z}\::\:\:{b}\:=\:{ka}\:……\left({i}\right) \\ $$$${b}\mid\:{c}\:\Leftrightarrow\:\exists\:{m}\:\in\mathbb{Z}\::\:\:{c}\:=\:{bm}…..\left({ii}\right) \\ $$$${eqn}\:\left({i}\right)\:{in}\:\left({ii}\right)\:\Rightarrow \\ $$$$\:\:{c}\:=\:\left({ka}\right){m} \\ $$$$\:\:{c}\:=\:{km}\:{a}\:\: \\ $$$${if}\:{k},{m}\in\:\mathbb{Z}\:{the}\:{km}\in\mathbb{Z} \\ $$$${hence}\:{a}\mid{c} \\ $$$$ \\ $$

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