Found problems: 310
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
There are $200$ numbers on a blackboard:
$ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$.
Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?
Let $n$ be a positive integer, and let $p$ be a prime number. Prove that if $p^p | n!$, then $p^{p+1} | n!$.
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
$\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$
What is $10! - 7! \cdot 6!$?
$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.
What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?
For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$?
$ \textbf{(A)}\ 8\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 21$
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$
Determine all positive integers $a$, $b$ and $c$ which satisfy the equation
$$a^2+b^2+1=c!.$$
[i]Proposed by Nikola Velov[/i]
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]
[i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.
Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $
[b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $
[i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $
[i]Cătălin Țigăeru[/i]
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?
Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
Given that
\[ \frac{((3!)!)!}{3!} = k \cdot n!, \]
where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n$.
For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation?
$$5! \cdot 9! = 12 \cdot N!$$
$\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$
Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?
A positive integer $n$ is called [i]cute[/i] when there is a positive integer $m$ such that $m!$ ends in exactly $n$ zeros.
a) Determine if $2019$ is cute.
b) How many positive integers less than $2019$ are cute?
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end?
(With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).