P is a natural number of at least $6$ digits and its left most digit is $7$. When this leftmost digit is removed from P, the number thus obtained is found to be $\frac{1}{21}$ times of P. What is the product of the all the non-zero digits of P?

$X!$ is completely divisible by ${11}^{51}$ but not by ${11}^{52}$. What is the sum of digits of largest such number X?

How many times would $1$ be used while writing all the natural numbers from $8$ to $127$ in the Binary number system?

What is the maximum number of elements that one can pick from the set of natural numbers from $1$ to $20$ such that the product of no two of them results in a perfect square or perfect cube?

$abcdefghij$ is a ten-digit number with distinct digits such that $a > b > c, d > e > f, g > h > i > j$. $a, b, c$ are consecutive even digits and $g, h, i, j$ are consecutive odd digits.If $d + e + f = 9$, then what is the value of $\left[\frac{a \times b \times c \times d}{i}\right]$ where [ ] denotes greatest integer function?