What is the first non zero digit in 50 factorial (50!)? What is the first non zero digit in 50 factorial (50!)?
Any Help or hint will be appreciated.

First, the exponent of $2$ in $50!$ is $47$, and the exponent of $5$ in $50!$ is $12$, so we’re looking for $\frac{50!}{10^{12}}$ mod$ 10$.

Since $\frac{50!}{10^{12}}$ is even (in fact, divisible by $2^{35}$), it suffices to compute it mod $5$:
$\frac{50!}{5^{12}}$ $\equiv$ ($4!$. $1$. $4!$. $2$. $4!$. $3$. $4!$. $4$. $4!$. $1$.($4!$. $1$. $4!$. $2$. $4!$. $3$. $4!$. $4$. $4!$). $2$
=$4!^{10}$.$4!^2$.2!
$\equiv$$(-1)^{10}$.$(-1)^2$.2
$\equiv$ $2$ mod $5$
$\frac{50}{10^{12}}$ $\equiv$ $2$.$2^{-12}$ $\equiv$ $2$ . $(2^4)^{-3}$ $\equiv$ $2$ mod $5$

So, $\frac{50!}{5^{12}}$ $\equiv$ $2$ (mod $10$ ) 
