Trailing zeroes

Can it be shown that no integer $n$ has an $n!$ with exactly 2015 trailing zeroes?

#NumberTheory

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Trailing zeroes of $n!$ is given by $z_n = \displaystyle \sum_{k=1}^\infty {\left \lfloor \frac{n}{5^k} \right \rfloor}$ , where $\lfloor x \rfloor$ is the greatest integer function. Therefore, there is a unique set of $z_n$ . The values of $z_n$ around $2015$ are as follows:

\[\begin{array} {} n: & 8055 & 8060 & 8065 & 8070 & 8075 & 8080 & 8085 & 8090 & 8095 \\ z_n: & 2011 & 2012 & 2013 & \color{red}{2014} & \color{red}{2016} & 2017 & 2018 & 2019 & 2020 \end{array} \]

It is seen that $2015$ is not a $z_n$ .

Chew-Seong Cheong - 5 years, 11 months ago

There is no need to construct the table. You should show that the lower bound of $n$ satisfy the equation $n\left(\frac15 +\frac1{5^2}+ \frac1{5^3}+\ldots\right) = 2015$ . Then you don't need to construct the table anymore and you're left to show that every increment by 5 of $n$ increases the trailing zeros by 1, and that the trailing zeros further increases by 1 when $n$ is a multiple of 25. Applying the formula, $z_{8060} = 2012$ , so $z_{8065} = 2012 + 1 = 2013$ , $z_{8070} = 2013 + 1 =2014 < 2015$ , $z_{8075} = 2014 + 1 + 1 = 2016 > 2015$ .

Pi Han Goh - 5 years, 11 months ago

Thanks, Pi Han. I was using a spreadsheet, the table comes to me easy. I was showing how $z_n$ changes with $n$ .

Chew-Seong Cheong - 5 years, 11 months ago

What is the proof of trailing zeroes?

Jåy Dïâz - 5 years, 3 months ago

Read trailing number of zeros

Pi Han Goh - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$