# Tagged Questions

Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### finding the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$ [closed]

What is the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$? How will I solve this type of problems?
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### The number of zeros in the expansion of $n!$ in base $12$

During an interview last year I was asked the following question: How many zeros appear at the end of $n!$ in base $12$, where $n$ is a positive integer? I applied the known Legendre formula for ...
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### Is it true for $n > 2$ then there always exists a prime $\le n$ that does not divide $n$?

I was thinking of how to prove $\frac{n^n}{n!}$ is never an integer for $n > 2$. I think if I prove the above question, then this follows immediately.
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### Prove $\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$ [duplicate]

Prove $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ Proof by induction: true for $n=2$. Assume true for $n$ and see if $n+1$ is true. $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ ...
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### If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$

If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$ Well I was able to prove this intuitively, but what i need is a rigorous mathematical proof. I shall explain my ...
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### Does (9/2)! have a real answer or not? [duplicate]

The TI-84 says 52.342777 but other calculators says domain error.
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### $n!>a$, can we solve for $n$ in terms of $a$? [duplicate]

Can we explicitly solve $n$ in terms of $a$? Can we rewrite this inequality in the form of $n>f(a)$ without using $n!$ the factorial?
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### Big O for factorials

Hello I have trouble proving:$$(n+1)!\notin O(n!)$$ My first step is the following: $$(n+1)!-cn!\le0$$ Can you please help me with the next step?
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### What does an exclamation point raised to a power, with no preceding number, mean?

In the OEIS sequence A049210, I noticed an odd notation I haven't seen before: a(n) = (8*n-1)(!^8), n >= 1, a(0) = 1. What does the ...
392 views

### Is it possible to calculate $\int x! dx$ [closed]

Is it possible to calculate $\int x! dx$, if yes ,then how and if no ,then why not? This question came in my mind when, I solved some questions on integration. Until now I haven't got the right ...
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### Problem involving factorials (divisibility) [closed]

Show that, for every $n \in \Bbb N$, the following number is natural: $$\frac {(n!)!} {{n!}^{(n-1)!}}$$. I dont't know how to prove, as I tried to find a way including combinatorics.
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### Factorials and the Mod function

I was just playing with the factorial and the modulo function. I just observed this interesting property. I was using a calculator $$13!\equiv 13\times 12\pmod{169}\\ 17!\equiv 17\times 16\pmod{289}$$...
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### Trying to simplify an expression for an induction proof.

I got it down to $(k+2)!-1 + (k+1)((k+1)!)$ I am trying to get it to $(k+2)!-1$ but I guess I do not understand factorials enough to simplify this. I am also assuming I am doing the induction ...
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### Calculate sum of infinite series by solving a differential equation

Calculate the sum of the infinite series $$\sum_{n=0}^{\infty}\frac{1}{(3n)!}$$ by solving an aptly chosen differential equation. I know that one can solve a differential equation by assuming that ...