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Question Number 73553 by 01 last updated on 13/Nov/19

$$\mathrm{give}:\mathrm{a},\mathrm{b},\mathrm{c}>0$$$$\mathrm{if}\mathrm{a}\mid \mathrm{b}\mathrm{and}\mathrm{b}\mid \mathrm{c}$$$$\mathrm{prove}:\mathrm{a}\mid \mathrm{c}$$

Answered by Rio Michael last updated on 13/Nov/19

$$a\mid b\iff \mathrm{\exists }k\in \mathbb{Z}:b=ka......\left(i\right)$$$$b\mid c\iff \mathrm{\exists }m\in \mathbb{Z}:c=bm.....\left(ii\right)$$$$eqn\left(i\right)in\left(ii\right)\Rightarrow$$$$c=\left(ka\right)m$$$$c=kma$$$$ifk,m\in \mathbb{Z}thekm\in \mathbb{Z}$$$$hencea\mid c$$$$$$

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