Let $a, b, m$ and $n$ be natural numbers such that $a>1$ and $b>1$. If $a^m b^n = 144^{145}$, then the largest possible value of $n-m$ is

The sum of all possible values of $x$ satisfying the equation $2^{4x^2} - 2^{2x^2+x+16} + 2^{2x+30} = 0$, is

For any natural numbers $m, n$, and $k$, such that $k$ divides both $m+2n$ and $3m+4n$, $k$ must be a common divisor of

For any non-zero real numbers $x, y$ such that $y \neq 3$ and $xy < x+3y-3$, will satisfy the condition

For some positive real number $x$, if $\log_3\sqrt{x} + \log_x(25)\cdot\log_x(0.008) = \dfrac{16}{3}$, then the value of $\log_3(3x^2)$ is

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

Let $k$ be the largest integer such that the equation $(x-1)^2 + 2kx + 11 = 0$ has no real roots. If $y$ is a positive real number, then the least possible value of $\dfrac{k}{4y} + 9y$ is

In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is

Minu purchases a pair of sunglasses at Rs.1000 and sells to Kanu at 20% profit. Then, Kanu sells it back to Minu at 20% loss. Finally, Minu sells the same pair of sunglasses to Tanu. If the total profit made by Minu from all her transactions is Rs.500, then the percentage of profit made by Minu when she sold the pair of sunglasses to Tanu is

Pipes A and C are fill pipes while Pipe B is a drain pipe of a tank. Pipe B empties the full tank in one hour less than the time taken by Pipe A to fill the empty tank. When pipes A, B and C are turned on together, the empty tank is filled in two hours. If pipes B and C are turned on together when the tank is empty and Pipe B is turned off after one hour, then Pipe C takes another one hour and 15 minutes to fill the remaining tank. If Pipe A can fill the empty tank in less than five hours, then the time taken, in minutes, by Pipe C to fill the empty tank is