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

For $n \in N$,the largest positive integer that divides $81^n + 20n - 1$ is $k$. If $S$ is the sum of all positive divisors of $k$,then $S - k =$

If $P$ is the greatest divisor of $49^{n}+16n-1$ for all $n \in N$,then the number of factors of $P$ is

The remainder when $64^{32^{32}}$ is divided by $9$ is equal to .........................

For any integer $n \geq 1$,the remainder when the expression $n^5-5n^3+4n+139$ is divided by $120$ is

The expression $x(x^{n-1} - na^{n-1}) + a^n(n-1)$ is divisible by $(x-a)^2$ for: