Let n and m be two positive integers such that there are exactly 41 integers greater than 8^m and less than 8^n, which can be expressed as powers of 2. Then, the smallest possible value of n + m is?

For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b² − 15)y = 8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is?

If x is a positive real number such that x⁸ + (1/x)⁸ = 47, then the value of x⁹ + (1/x)⁹ is

For a real number x, if 1/2, log₃(2ˣ − 9)/log₃4, and log₅(2ˣ + 17/2)/log₅4 are in arithmetic progression, then the common difference is

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is:

A quadratic equation x² + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

Let n be any natural number such that 5ⁿ⁻¹ < 3ⁿ⁺¹. Then, the least integer value of m that satisfies 3ⁿ⁺¹ < 2ⁿ⁺ᵐ for each such n, is?

A merchant purchases a cloth at a rate of Rs.100 per meter and receives 5 cm length of cloth free for every 100 cm length of cloth purchased by him. He sells the same cloth at a rate of Rs.110 per meter but cheats his customers by giving 95 cm length of cloth for every 100 cm len…

A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, ta…