For which value of n $\in$N, n! has 13 trailing zero

n! has 13 trailing zeros $\Rightarrow$Index of 10 in n! is 13 $\Rightarrow$Index of in n! is 13.

Index of 5 in 51! is $\left[\frac{51}{5}\right]+\left[\frac{51}{5^2}\right]+\left[\frac{51}{5^3}\right]+\dots =10+2+0\dots =12$

Index of 5 in 54! is $\left[\frac{54}{5}\right]+\left[\frac{54}{5^2}\right]+\left[\frac{54}{5^3}\right]+\dots =10+2+0\dots =12$

Index of 5 in 57! is $\left[\frac{57}{5}\right]+\left[\frac{57}{5^2}\right]+\left[\frac{57}{5^3}\right]+\dots =11+2+0\dots =13$

Index of 5 in 60! is $\left[\frac{60}{5}\right]+\left[\frac{60}{5^2}\right]+\left[\frac{60}{5^3}\right]+\dots =12+2+0\dots =14$