The number of positive divisors of $252$ is

If the integer represented by $100!$ has $K$ consecutive zeroes at the end,then $K=$

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 $n$ is a factor of $72$,such that $xy = n$,then the number of ordered pairs $(x, y)$ is: (where $x, y \in N$)

The coefficient of $x^7$ in $(1-x+2x^3)^{10}$ is $........$.

The largest non-negative integer $k$ such that $24^k$ divides $13!$ is