<s>[INST] 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.
 [/INST] 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 $10$ ) 
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