# Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### $\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ [duplicate]

Show that $\binom{p^a-1}{k}\equiv(-1)^k\pmod{p}$ for $0\le k\le p^{a}-1$. Solution: $\binom{p^a-1}{k}=\frac{(p^a-1)(p^a-2)\cdots(p^a-k)}{k!}$. Idea: Then power of $p$ in $k!$ must be less than ...
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### What is the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$?

Without using computer programs, can we find the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$? What I know is that the last non-zero ...
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### a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
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### To prove that $(n-1)!+1$ is not a power of $n$.

If $n$ is composite, prove that $(n-1)!+1$ is not a power of $n$. Hint: We know that if $n$ is composite and $n>4$ then $(n-1)!+1$ is divisible by $n$. My Solution: Since $n=4$ is the first ...
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### Find all pairs $(k, n)$ of positive integers such that $k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$

Find all pairs $(k, n)$ of positive integers such that $$k! = (2^n − 1)(2^n − 2)(2^n − 4) · · · (2^n − 2^{n−1})$$ I tried to solve this problem but only found one solution $(1,1)$. Please help me to ...