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Question Number 112813 by Aina Samuel Temidayo last updated on 09/Sep/20

A binary operation has the property  a∗(b∗c) = (a∗b)•c and that a∗a=1 for  all non−zero real numbers a,b and c.  (′•′ here represent multiplication).  The solution of the equation  2016∗(6∗x)=100 can be written as (p/q)  where p and q are relatively prime  positive integers. What is q−p?

$$\mathrm{A}\:\mathrm{binary}\:\mathrm{operation}\:\mathrm{has}\:\mathrm{the}\:\mathrm{property} \\ $$$$\mathrm{a}\ast\left(\mathrm{b}\ast\mathrm{c}\right)\:=\:\left(\mathrm{a}\ast\mathrm{b}\right)\bullet\mathrm{c}\:\mathrm{and}\:\mathrm{that}\:\mathrm{a}\ast\mathrm{a}=\mathrm{1}\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{non}−\mathrm{zero}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}. \\ $$$$\left('\bullet'\:\mathrm{here}\:\mathrm{represent}\:\mathrm{multiplication}\right). \\ $$$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{2016}\ast\left(\mathrm{6}\ast\mathrm{x}\right)=\mathrm{100}\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\frac{\mathrm{p}}{\mathrm{q}} \\ $$$$\mathrm{where}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{are}\:\mathrm{relatively}\:\mathrm{prime} \\ $$$$\mathrm{positive}\:\mathrm{integers}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{q}−\mathrm{p}? \\ $$

Commented by kaivan.ahmadi last updated on 09/Sep/20

  2016∗(6∗x)=(2016∗6)•x=100⇒  (((2016)/6)•6∗6)•x=100⇒(((2016)/6)•1)•x=100⇒  ((2016)/6)•x=100⇒x=100•(6/(2016))=((600)/(2016))=((25)/(84))  (25,84)=1⇒p=25,q=84⇒q−p=59

$$ \\ $$$$\mathrm{2016}\ast\left(\mathrm{6}\ast{x}\right)=\left(\mathrm{2016}\ast\mathrm{6}\right)\bullet{x}=\mathrm{100}\Rightarrow \\ $$$$\left(\frac{\mathrm{2016}}{\mathrm{6}}\bullet\mathrm{6}\ast\mathrm{6}\right)\bullet{x}=\mathrm{100}\Rightarrow\left(\frac{\mathrm{2016}}{\mathrm{6}}\bullet\mathrm{1}\right)\bullet{x}=\mathrm{100}\Rightarrow \\ $$$$\frac{\mathrm{2016}}{\mathrm{6}}\bullet{x}=\mathrm{100}\Rightarrow{x}=\mathrm{100}\bullet\frac{\mathrm{6}}{\mathrm{2016}}=\frac{\mathrm{600}}{\mathrm{2016}}=\frac{\mathrm{25}}{\mathrm{84}} \\ $$$$\left(\mathrm{25},\mathrm{84}\right)=\mathrm{1}\Rightarrow{p}=\mathrm{25},{q}=\mathrm{84}\Rightarrow{q}−{p}=\mathrm{59} \\ $$$$ \\ $$

Commented by Aina Samuel Temidayo last updated on 09/Sep/20

Thanks.

$$\mathrm{Thanks}. \\ $$

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