Trailing zeroes and inequalities?

Number Theory Level 3

Define the function $\psi (n)$ as the number of trailing zeroes in $n!$ .

What is the supremum value of $\frac{\psi (n)}{n}$ ?

1 solution

Md Zuhair
Jun 8, 2017

Here the number is $n$ .

Now max of $(\psi{n})$

By defination, we know

$\dfrac{n}{5} + \dfrac{n}{25} + \dfrac{n}{125}+... \infty$

$\implies \dfrac{\dfrac{n}{5}}{1-\dfrac{1}{5}}$

$\implies \dfrac{\dfrac{n}{5}}{\dfrac{4}{5}}$

$\implies \dfrac{n}{\color{#D61F06}\boxed{4}}$

Answer: 4