The fractional part of the number $\frac{4^{2022}}{15}$ is equal to

If $3^{2n+2}-8n-9$ is divisible by $2^p$ for all $n \in N$,then the maximum value of $p$ is

The remainder when $10^{10} \cdot (10^{10} + 1) \cdot (10^{10} + 2)$ is divided by $6$ is

The value of the greatest positive integer $k$,such that $49^k + 1$ is a factor of $48(49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1)$ is

The remainder when $7^n - 6n - 50$ $(n \in N)$ is divided by $36$ is:

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