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Francisco G.

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Number Theory

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Question:

What is the least number $$n$$ sucht that $$n!$$ ends with at least 100 zeros?

Francisco G.

Answer:

For any number $$n$$ we have that $$n!$$ ends with $$ N=\left\lfloor\frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{5^2}\right\rfloor +\cdots+\left\lfloor\frac{n}{5^k}\right\rfloor+\cdots $$ zeros, where $$\lfloor \cdot \rfloor$$ denote the integer part. Note that $$ \left\lfloor\frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{25}\right\rfloor\le N, $$ as we search for a value of $$n$$ such that $$N$$ is near of 100, we have that $$n/5+n/25\approx 100$$, therefore $$6n\approx 2500$$ and $$n\approx 416$$. If $$n=416$$ we have $$ N=\left\lfloor\frac{416}{5}\right\rfloor+\left\lfloor\frac{416}{5^2}\right\rfloor+\left\lfloor\frac{416}{5^3}\right\rfloor+\left\lfloor\frac{416}{5^4}\right\rfloor+\cdots=83+16+3+0=102. $$ Now, $$415=83\times 5$$ and $$410=82\times 5$$, therefore $$409!$$ ends with $$102-2=100$$ zeros. Note that $$405!$$ also ends with $$100$$ zeros and $$404!$$ ends with at most $$99$$ zeros because $$5$$ divides $$405$$, so $$405$$ is the least number with that property.

LaTeX

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Question:

How to define a new math environment (lemma, corollary, etc) in Latex?

Francisco G.

Answer:

To define a new environment we can use the following structure: \usepackage{amsthm} \newtheorem{NewEnv}{New Environment} Example of use \begin{NewEnv} ...Text... \end{NewEnv}

Calculus

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Question:

Show that the function $$ F(x)=\int_2^x \frac{1}{\log t}dt $$ is not bounded in $[2, \infty)$.

Francisco G.

Answer:

As $$\log$$ is an increasing function we have $$\log t\le \log x$$ for all $$t\in [2,x]$$, then $$ F(x)=\int_2^x \frac{1}{\log t}dt\ge \int_2^x \frac{1}{\log x}dt=\frac{x-2}{\log x}. $$ But, by the L'Hôpital rule $$ \lim_{x\to \infty} \frac{x-2}{\log x}=\lim_{x\to \infty}\frac{1}{1/x}=\lim_{x\to\infty} x=\infty. $$ Therefore, $$ \lim_{x\to\infty}F(x)=\infty $$ and it follows that $$F$$ is not bounded.

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