# Tagged Questions

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### Prove that $6! \mid n(n+1)…(n+5)$ [closed]

Prove that for all $n \in \mathbb{Z}$, $6! \mid n\cdot(n+1)\cdots(n+5)$ using only criteria of divisibility (without using combinatorial arguments).
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### Factorial Summation Problem [duplicate]

$$\sum_{j=0}^n j\cdot j!$$ I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
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### Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
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### Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
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### Even or Odd for factorial

Moderator Note: This is a current contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this ...
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### What are the conditions for $n^2 \nmid(n-1)!$

Q: What are the conditions for $n^2 \nmid (n-1)!$, given that $2\le n \le 100$ and $n\in \mathbb{N}$? According to me the two conditions must be: 1. $n$ is a prime number (since the factorization ...
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### How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $2^{30}3^{13}5^5 7^3$. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
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### Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
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### What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I tried to do it using binomial theorem but that doesn't help. How will we do this? Please help.
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### Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
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### Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
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### Show that $p!$ and $(p - 1)! - 1$ are relatively prime

If $p$ is prime number, with $p>3$ Show that $p!$ and $(p - 1)! - 1$ are relatively prime. I tried $\text{gcd}\;(p!,(p-1)!-1)=d\Longrightarrow d\mid p!$ e $d\mid(p-1)!-1$ having ...
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show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $$\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$$ I have a theorem (shown in my text) that says $$\fbox{If a_1,...,a_m\in\mathbb{N}^* then ... 2answers 101 views ### Show that there is no natural number n such that 3^7 is the largest power of 3 dividing n! Show that there is no natural number n such that 7 is the largest power a of 3 for which 3^a divides n! After doing some research, I could not understand how to start or what to do to ... 3answers 76 views ### If N is a multiple of 100, N! ends with \left(\frac{N}4-1 \right) zeroes. Did certain questions about factorials, and one of them got a reply very interesting that someone told me that it is possible to show that If N is a multiple of 100, N! ends with ... 1answer 39 views ### Find the smallest value of n so that the greater potency of 5 which divides n! is 5^{84}. What are the other numbers that enjoy this property? Find the smallest value of n so that the greater potency of 5 which divides n! is 5^{84}. What are the other numbers that enjoy this property? I thought I would put together an equation ... 2answers 85 views ### Find the greatest power of 104 which divides 10000! Find the greatest power of 104 which divides 10000! I thought$$104=2^3\cdot13$$so I have to find n such that$$(2^3\cdot13)^n\mid 10000!$$Obviously, we can see that there are fewer ... 1answer 99 views ### Find the prime factor decomposition of 100! and determine how many zeros terminates the representation of that number. Find the prime factor decomposition of 100! and determine how many zeros terminates the representation of that number. Actually, I know a way to solve this, but even if it is very large and ... 2answers 114 views ### Show that \frac{(2n)!}{(n)!}=2^n(2n-1)!! Show that \frac{(2n)!}{(n)!}=2^n(2n-1)!! is the question I am struggling with. I started by saying: (2n)!=2n(2n-1)(2n-2)(2n-3)...3*2*1 But then I'm stuck. 1answer 58 views ### Congruences with prime number and factorial Prove that if p\equiv 1 \pmod{4} is a prime number and$$x\equiv \pm \left(\frac{p-1}{2}\right)! \pmod{p}$$then x^2\equiv -1 \pmod{p} I think Wilson's theorem will come in handy here, used ... 2answers 1k views ### the nth root of n!? I am playing around with the root/ratio test to practice with series. I just showed that \sum \frac{1}{n!} converges by using the ratio test. I decided to see how things would go with the root test ... 3answers 159 views ### Integer ordered pairs (x,y) for which x^2-y!… [1] Total no. of Integer ordered pairs (x,y) for which x^2-y! = 2001 [2] Total no. of Integer ordered pairs (x,y) for which x^2-y! = 2013 My Try:: (1) x^2-y! = 2001\Rightarrow x^2 = ... 1answer 274 views ### How is it possible that \infty!=\sqrt{2\pi}? I read from here that:$$\infty!=\sqrt{2\pi}$$How is this possible ?$$\infty!=1\times2\times3\times4\times5\times\ldotsBut \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ ... 1answer 99 views ### The number of zeros in the decimal representation of the factorial of 126 How many zeros are in 126! ... the result is 34. But can I calculate it manually? I have seen How many zeroes are in 100! but I don't think it's helpful. 12answers 2k views ### Can the factorial function be written as a sum? I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist. 3answers 126 views ### Product representations of the factorial function? Is this the only product representation of the factorial function? {n!} =\prod_{k=1}^{n} k $$2answers 703 views ### how to find remainder when 20! + 20^{23} is divided by 23? how to find remainder when 20! + 20^{23} is divided by 23? I am finding it bit difficult to solve. Does any one has a simpler way to solve this problem?? 2answers 177 views ### Polynomials mapping factorials to factorials I'm looking for all polynomials P(x) with integer coefficients such that for every n \in \Bbb N there is an m \in \Bbb N such that P(n!)=m!. The only solutions seem to be the constant ... 0answers 66 views ### When is n!+1 a square? [duplicate] I'm looking for the solutions (n,m) of the equation n!+1=m^2. I have calculated the values of \sqrt{n!+1} for n \le  and found only the solutions (4,5), (5,11) and (7,71). Are these ... 0answers 65 views ### Simplify \frac{[m+n-1]!}{[m]![n]!} where [k]=x^k-x^{-k} and [k]!=[2][3]…[k]. Adopting the notation [k] = x^k - x^{-k}  and [k]! = [2][3]...[k] (note that [1] is omitted), and letting m,n be two integers greater than 1 such that n>m and gcd(m,n)=1, would it be ... 2answers 213 views ### Factorials and Divisibility I'm having trouble getting started on the following: Given n_1, n_2, ..., n_k \in \Bbb N, show that n_1!\cdot n_2!\cdot\cdot\cdot n_k! |(n_1+n_2+...+ n_k)! I thought about a proof by ... 1answer 180 views ### Understanding the upper and lower bounds of the error estimate in Stirling's Approximation Based on the Wikipedia article on Stirling Approximation: n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n} where \frac{1}{12n+1} < \lambda_n < \frac{1}{12n} How would this ... 2answers 238 views ### Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation I am trying to get the lower bound for: f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ... 1answer 187 views ### Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function? In Ramanujan's proof of Bertrand's Postulate, Ramanujan states: \log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x) where: \vartheta(x) = \sum_{p \le x} ... 0answers 436 views ### Understanding Ramanujan's approach in his proof of Bertrand's Postulate I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of \varphi(2x) - \varphi(x) What would be wrong with this approach for ... 3answers 62 views ### Constrictions on A.P with factorials. There are five numbers (a_1,a_2,a_3,a_4,a_5), such that they are in Arithmetic Progression. Given that a_1 and a_2 are factorials, is there a possibility that either a_4 OR a_5 is a ... 2answers 190 views ### Factorials and Arithmetic Progression. Are there sets of factorials (a_1!,a_2!,a_3!,\dots,a_n!), such that they exist in Arithmetic progression. n is a natural number I don't see any such examples(Except for n=2). And I don't see ... 1answer 139 views ### Solving n!+m!+k^2=n!m! for positive integers n,m,k I have been running in circles with this for a while now. It seems that the only solution is (n,m,k)=(2,3,2) but I don't know how to prove it. Things I have noticed: WLOG n\geq m we see that ... 3answers 314 views ### Factorial expressed in terms of two other factorials Can the factorial of N always be expressed by the sum(addition and subtraction) or the product of two other factorials? Do there always exist integer A and B such that N! = A! + B!, or N! = ... 2answers 2k views ### Number of zero digits in factorials Here is a riddle someone has been asked in a job interview: How many zero digits are there in 100!? Well, I found the first 24 quite fast by counting how many times five divides 100! (5 ... 2answers 253 views ### Prove quotient of factorials is integral If n is an integer \gt 0, prove$$\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$$is also an integer. I understand that a general approach is proving that the power of any prime factor is greater in the ... 1answer 229 views ### Need help with Factorial Sums! [duplicate] Possible Duplicate: How to prove that the number 1!+2!+3!+\dots+n! is never square? Show that the sum$$\sum_{k=1}^nk!\neq m^2$$for any integer m, for n\geq4. 3answers 263 views ### When is a factorial of a number equal to its triangular number? Consider the set of all natural numbers n for which the following proposition is true.$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$Here's an example:$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
Without using a calculator, how can we solve the following? How do we find the number of zeros at the end of $600!$ What are the last 3-digits of $171^{172}$? What is the sum of all positive numbers ...
### Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$
Note: this is a programming challenge at this site For this equation $$\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}\quad ( N \text{ factorial} ),$$ find the number of positive integral solutions for ...