# [SOLVED]prove for all Z+

#### karush

##### Well-known member
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

##### Well-known member
$a \mid b$ 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

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

#### karush

##### Well-known member
ok i have never tried that
are you sure the vertical bar means a/b?

#### topsquark

##### Well-known member
MHB Math Helper
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

#### karush

##### Well-known member
oh
can we prove this just by setting n=1

#### MountEvariste

##### Well-known member
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 $120$.

#### topsquark

##### Well-known member
MHB Math Helper
oh
can we prove this just by setting n=1
From there use Mathematical Induction. It should be rather easy but I haven't really spent much time thinking about it. (ie. If it works for n = 1 then you always have that factor.)

-Dan