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Question Number 143630 by Eric002 last updated on 16/Jun/21
prove that if a and c are odd integers  then ab+bc is even for every integer b?
$${prove}\:{that}\:{if}\:{a}\:{and}\:{c}\:{are}\:{odd}\:{integers} \\ $$$${then}\:{ab}+{bc}\:{is}\:{even}\:{for}\:{every}\:{integer}\:{b}? \\ $$
Commented by mr W last updated on 16/Jun/21
yes.  a,c=add  ⇒a+c=even  even×any integer=even  ⇒(a+c)b=even
$${yes}. \\ $$$${a},{c}={add} \\ $$$$\Rightarrow{a}+{c}={even} \\ $$$${even}×{any}\:{integer}={even} \\ $$$$\Rightarrow\left({a}+{c}\right){b}={even} \\ $$
Answered by Rasheed.Sindhi last updated on 16/Jun/21
Let a=2m+1 & c=2n+1  ab+bc=b(a+c)=b(2m+1+2n+1)           =b(2m+2n+2)           =2b(m+n+1) (Even)
$${Let}\:{a}=\mathrm{2}{m}+\mathrm{1}\:\&\:{c}=\mathrm{2}{n}+\mathrm{1} \\ $$$${ab}+{bc}={b}\left({a}+{c}\right)={b}\left(\mathrm{2}{m}+\mathrm{1}+\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:={b}\left(\mathrm{2}{m}+\mathrm{2}{n}+\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:=\mathrm{2}{b}\left({m}+{n}+\mathrm{1}\right)\:\left(\mathcal{E}{ven}\right) \\ $$

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