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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