$\displaystyle \binom{5^n}{i}$ isn't divisible by $\displaystyle 5^n$ for $\displaystyle i=5^n$. But anyway, the congruence is mod $\displaystyle 5^n+1$.
$\displaystyle \binom{5^n}{i}$ isn't divisible by $\displaystyle 5^n$ for $\displaystyle i=5^n$. But anyway, the congruence is mod $\displaystyle 5^n+1$.
Of course: the sum should have been split into three parts: for $\displaystyle i=0\,,\,i=5^n\,,\,1\leq i\leq 5^n-1$