Found problems: 310
Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.
Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?
Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$
How many zeroes appear at the end of $209$ factorial?
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]
Find all pairs of positive integers $(m,n)$ such that
$$lcm(1,2,\dots,n)=m!$$
where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.
Find all pairs of positive integers \( n \) and \( x \) such that
\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]
[i](Petko Lazarov, Bulgaria)[/i]
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.
Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$
.
Consider all the possible subsets of the set $\{1,2,..., N\}$ which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is $(N + 1)! - 1$.
(Based on idea of R.P. Stanley)
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).
Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.
For each positive integer $n$ denote:
\[n!=1\cdot 2\cdot 3\dots n\]
Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.
Prove that you can represent an arbitrary number not exceeding $n!$ as a sum of $k$ different numbers ($k\le n$) that are divisors of $n!$.
In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end?
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$
What is the value of $\dfrac{11!-10!}{9!}$?
$\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
The number $2013$ is expressed in the form \[2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},\] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
Determine all pairs $(k, n)$ of positive integers that satisfy
$$1! + 2! + ... + k! = 1 + 2 + ... + n.$$
Prove that
$$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$
A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?
Let $0!!=1!!=1$ and $n!!=n\cdot (n-2)!!$ for all integers $n\geq 2$. Find all positive integers $n$ such that
\[\dfrac{(2^n+1)!!-1}{2^{n+1}}\]
is an integer.