The number of integer value(s) of n when `3^512`-1 is divided by `2^n` is/are

If n is a positive integer then 11^n-10n-1 is divided by

Let n be the value of the least integer (0 gt n gt 5) so that 3^(2n)+4 is not prime. What is the value of the remainder when 3^(2n)+4 is divided by n?

The number of values of n such that log_(n) 2^(231) is an integer is (n in N)

A certain number 'C' when divided by N_1 it leaves a remainder of 13 and when it is divided by N_2 it leaves a remainder of 1, where N_1 and N_2 are the positive integers. Then the value of N_1 + N_2 is if (N_1)/(N_2) = 5/4 .