If the constant term in the expansion of $\left(3 x^{3}-2 x^{2}+\frac{5}{x^{5}}\right)^{10}$ is $2^{k} \cdot l$,where $l$ is an odd integer,then the value of $k$ is equal to

The number of positive integers $n$ such that $n+3$ divides $n^3-3$ is

Let $f(x) = 2x^n + \lambda$,where $\lambda \in R$ and $n \in N$. Given $f(4) = 133$ and $f(5) = 255$,find the sum of all the positive integer divisors of $(f(3) - f(2))$.

If $2^{n}$ divides $16!$ and $2^{n+1}$ does not divide $16!$,then $n=$

The arithmetic mean of five natural numbers is $40$. The largest exceeds the smallest number by $10$. If $\alpha$ is the maximum possible value for the largest of these $5$ numbers,then the number of positive integral divisors of $\alpha$ is

Let $n$ be a non-negative integer. Then the number of divisors of the form $4n+1$ of the number $(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$ is equal to $....$