If $n \in N$,then ${x^{2n - 1}} + {y^{2n - 1}}$ is divisible by

The number $(101)^{100}-1$ is divisible by

If the remainder when $x$ is divided by $4$ is $3$,then the remainder when $(2020+x)^{2022}$ is divided by $8$ is ....... .

If the greatest divisor of $30 \cdot 5^{2n} + 4 \cdot 2^{3n}$ is $p, \forall n \in N$ and that of $2^{2n+1} - 6n - 2$ is $q, \forall n \in N$, then $p+q=$

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

Statement $-1$: For every natural number $n$,$(n + 1)^7 - n^7 - 1$ is divisible by $7$.
Statement $-2$: For every natural number $n$,$n^7 - n$ is divisible by $7$.