Strange Algebra

Algebra Level 4

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

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

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

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

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

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