For what value of * the statement $$\left( {\frac{*}{{15}}} \right)$$ $$\left( {\frac{*}{{135}}} \right)$$  = 1 is true ?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Method 1:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow x = \sqrt {15 \times \left(15 \times 9 \right)} \cr & \Leftrightarrow x = \sqrt {{{15}^2} \times {3^2}} \cr & \Leftrightarrow x = 15 \times 3 \cr & \Leftrightarrow x = 45 \cr & \cr & {\text{Method 2:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow {x^2} = 2025 \cr & \Leftrightarrow x = \sqrt {{2025} } \cr & \Leftrightarrow x = 45 \cr} $$