The remainder when $((64)^{(64)})^{(64)}$ is divided by $7$ is equal to

If $n \in N$,then ${7^{2n}} + {2^{3n - 3}} \cdot {3^{n - 1}}$ is always divisible by

Using the binomial theorem,prove that $6^{n} - 5n$ always leaves a remainder of $1$ when divided by $25$ for all $n \in N$.

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

Let $n > 1$ be an integer. Which of the following sets of numbers necessarily contains a multiple of $3$?

The $50^{\text{th}}$ root of a number $x$ is $12$ and the $50^{\text{th}}$ root of another number $y$ is $18$. Then the remainder obtained on dividing $(x + y)$ by $25$ is $........$.