# Prove for all Z+

• MHB
Gold Member
MHB
prove
$10^{\dfrac{5^n -1}{4}} \Huge\vert (5^n)!$
for all $Z^+$

ok this was sent to me on email but thot I could solve it but ?

first I assume the vertical bar means such that

not sure if ! means factorial or not negative

anyway curious

MountEvariste
$a \mid b$ means $a$ divides $b$.

The following formula may help.

Legendre's formula: For any prime number $p$ and any positive integer $m$, let
${\displaystyle \nu _{p}(m)}$ be the exponent of the largest power of $p$ that divides $m$. Then

$${\displaystyle \nu _{p}(m!)=\sum _{i=1}^{\infty }\left\lfloor {\frac {m}{p^{i}}}\right\rfloor}$$

Gold Member
MHB
ok i have never tried that
are you sure the vertical bar means a/b?

Gold Member
MHB
ok i have never tried that
are you sure the vertical bar means a/b?
Not a/b it's a divides into b evenly. For example, 3|15 and 5|15.

-Dan

Gold Member
MHB
oh
can we prove this just by setting n=1

MountEvariste
oh
can we prove this just by setting n=1
I don't know what you mean.

It's true for $n=1$ because $10$ divides $120$.