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For all integers a,b,c, prove or disprove each of the following:

i. If a | c and b | c, then ab | c^2.

ii. If a | (b+c), then a | b and a | c.

Note : "If true prove without using numbers as an example otherwise you can use them if false" That's What My Prof. says.

Unfortunately, The first one I couldn't solve it.

Note No. 2 : My Question is different than this Question : $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$

Please help me , I didn't understand the question really

Thank you in advance

Jim Henry
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1 Answers1

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Your question is not different than the one you mentioned, because your question is a particular case of it, with $b=d$.

Scientifica
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  • I don't know how to extract what is equivalent to my case really. @Scientifica – Jim Henry Apr 06 '20 at 22:33
  • I mentioned it because someone hide my question. @Scientifica – Jim Henry Apr 06 '20 at 22:35
  • I ask this question because it is difficult for me , The person (who closed or hide my question I don't know what happened) mentioned The linked Question that is complicated for me ) so how can I understand the second Question If I didn't understand the first one ?? @Scientifica – Jim Henry Apr 06 '20 at 22:44
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    @JimHenry Let's use colors then... if $\color{blue}{a}\mid \color{green}{c}$ and $\color{red}{b}\mid \color{purple}{c}$ then $\color{blue}{a}\color{red}{b}\mid \color{green}{c}\color{purple}{c}$. Just as how if $\color{blue}{a}\mid \color{green}{b}$ and $\color{red}{c}\mid \color{purple}{d}$ then $\color{blue}{a}\color{red}{c}\mid \color{green}{b}\color{purple}{d}$ – JMoravitz Apr 06 '20 at 22:48
  • "It is too complicated for me" I'm sorry you feel that way, but to continue you really must understand this proof and others like it. This is just applying the definitions... as I suggested in the previous time you asked this, maybe you should use a more tactile definition of divisibility utilizing the definition that $a\mid c$ if and only if there exists some integer $n$ such that $c = n\cdot a$. Note that if $c=na$ as well as $c=mb$ for some integers $m,n$ then what does $c^2$ equal? Is it an integer multiple of $ab$? – JMoravitz Apr 06 '20 at 22:51
  • Thank you very much, I understand it now @JMoravitz – Jim Henry Apr 06 '20 at 22:52
  • @JimHenry I just came in and saw JMoravitz' great answers with colors, as well as good explanation of the proof. Glad you understood. – Scientifica Apr 06 '20 at 22:53
  • Okay, I will practice it again , Thanks you very much:) @JMoravitz – Jim Henry Apr 06 '20 at 22:58