Strange Algebra

Algebra Level 4

Let S\color {Red}{'S'} be the ordered pairs of (x,y)\color{Blue}{(x,y)} such that 0<x1\color{Green}{0<x\le 1} and 0<y1 \color{Purple}{0<y\le 1} and [ log3(1x) ]\color{Brown}{ [\ log _3(\frac{1}{x})\ ]}and [ log5(1y) ]\color{DarkRed}{[ \ log _5(\frac{1}{y}) \ ]} are both even.

While plotting a graph between the values of x,y\color{DarkRed}{x,y} that satisfy the condition the area of the graph comes out to be of the form mn\color{Maroon}{\frac{m}{n}} where m\color{Magenta}{m} and n\color{magenta}{n} are co-prime integers.

Then what is the value of m+n?\color{Green}{m+n ?} NOTE\color{Red}{NOTE}

  • [.]\color{Magenta}{[ .]} represents GIF\color{Magenta}{GIF}

For eg. - [1.2]=1 , [2.3]=2


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