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

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

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

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

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

$$\mathrm{Thanks}.$$

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