If $a, b$ and $n$ are natural numbers,then $a^{2n-1} + b^{2n-1}$ is divisible by

If $X = \{ 8^n - 7n - 1 : n \in N \}$ and $Y = \{ 49(n - 1) : n \in N \}$,then

The expression $25^{190} - 19^{190} - 8^{190} + 2^{190}$ is divisible by:

If $(27)^{999}$ is divided by $7$,then the remainder is

If $[x]$ denotes the greatest integer function of $x$,then the number of positive integral divisors of $\left[(2+\sqrt{3})^5\right]$ is

The remainder when $7^{7^{7^{...7}}}$ ($22$ times $7$) is divided by $48$ is