<s>[INST] Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$ 
Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$

Attempt:
I want to use the following theorem:

The largest exponent of $e$ of a prime $p$ such that $p^e$ is a divisor of $n!$ is given by $$e=[\frac{n}{p}]+[\frac{n}{p^2}]+[\frac{n}{p^3}]+\cdots$$.

The number of times the prime divisor $5$ is repeated in $(5^{25}-1)!$ equals the greatest exponent of $5$ contained in $(5^{25}-1)!$, which is  $e_1=[\frac{(5^{25}-1)}{5}]+[\frac{(5^{25}-1)}{5^2}]+[\frac{(5^{25}-1)}{5^3}]+[\frac{(5^{25}-1)}{5^4}]+\cdots$
The number of times the prime divisor $2$ is repeated in $(5^{25}-1)!$ equals the greatest exponent of $2$ contained in $(5^{25}-1)!$, which is $e_2=[\frac{(5^{25}-1)}{2}]+[\frac{(5^{25}-1)}{2^2}]+[\frac{(5^{25}-1)}{2^3}+[\frac{(5^{25}-1)}{2^4}]+\cdots$
Therefore, the number of zeros at the right end equals the greatest exponent of $10$ contained in $(5^{25}-1)!$$=\min{(e_1, e_2)}$. 

I am unable to simplify $e_1, e_2$. Please help me. 

 [/INST] First in the définition of $e_1$ and $e_2$ there is no factoriel $!$ we have:
$$ e_1=\left\lfloor\frac{5^{25}-1}{5}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^2}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^3}\right\rfloor+\left\lfloor\frac{5^{25}-1}{5^4}\right\rfloor+\cdots+\left\lfloor\frac{{5^{25}-1}}{5^{24}}\right\rfloor
$$
And here you forgot to add the function : the greatest number less than $5^n-1$ each time because the numbers to sum must be all integers and finally we have to stop at $5^{24}$ because the rest are all null and we have:
$$e_1=5^{24}-1+5^{23}-1+\dots+5^0-1=\frac{5^{25}-1}{4}-25=\frac{5^{25}-101}{4}$$
and it's clear that the greatest power of $5$ dividing $(5^{25}-1)!$ is the smallest one. each time we have a $5$ n=in the factoriel we will have more then $3$ powers of $2$, you can prove this by comparing the formula of $e_1$ and $e_2$ for instance here is an example $1\cdot 2\cdot 3\cdot 4\cdot 5=5\cdot 3\cdot 2^3$
</s>