Divisibility of Fibonomial coefficients in terms of their digital representations and applications

We give a characterization for the integers $n\ge 1$ such that the Fibonomial coefficient ${(\genfrac{}{}{0ex}{}{pn}{n})}_F$ is divisible by $p$ for any prime $p\ne 2,5$. Then we use it to calculate asymptotic formulas for the number of positive integers $n\le x$ such that $p\mid {(\genfrac{}{}{0ex}{}{pn}{n})}_F$. This completes the study on this problem for all primes $p$.