Strange Algebra

Algebra Level 4

Let S\color{#D61F06}{'S'} be the ordered pairs of (x,y)\color{#3D99F6}{(x,y)} such that 0<x1\color{#20A900}{0<x\le 1} and 0<y1 \color{#69047E}{0<y\le 1} and [ log3(1x) ]\color{#624F41}{ [\ 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{#20A900}{m+n ?} NOTE\color{#D61F06}{NOTE}

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

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

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