The remainder when $428^{2024}$ is divided by $21$ is ............

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

Let the number $(22)^{2022} + (2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $(\alpha^2 + \beta^2)$ is equal to

If $P(n):$ " $2^{2n}-1$ is divisible by $k$ for all $n \in N$ " is true,then the value of $k$ is:

The remainder on dividing $1+3+3^{2}+3^{3}+\ldots+3^{2021}$ by $50$ is

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