If $2 \cdot 4^{2k+1} + 3^{3k+1} = 11t$ and $2 \cdot 4^{2k+3} + 3^{3k+4} = 11(pt + 3^q)$,where $k, t \in Z^{+}$,then $(p, q) =$

Let $a, b, c, d$ be positive integers. Consider the following statements:
$I$. If $9$ divides $a^3+b^3+c^3$,then $3$ divides $abc$.
$II$. If $9$ divides $a^3+b^3+c^3+d^3$,then $3$ divides $abcd$.

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

$x^n + y^n$ is divisible by

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

If $n \in N$,then ${11^{n + 2}} + {12^{2n + 1}}$ is divisible by