1
vote
3answers
245 views

Prove that no number in this list is prime - Formatting a proof advice

Question: Let $n \in \mathbb{Z}$ where $n \geq 2$, prove no number in the list: $$n! + 2, n! + 3,...,n! + n$$ is prime. I have written my proof exactly as follows: Proof: $P(n) = n! + n = ...
-2
votes
2answers
99 views

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 ...
8
votes
1answer
113 views

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 ...
3
votes
2answers
424 views

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 ...
6
votes
2answers
127 views

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 $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
6
votes
2answers
359 views

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.
4
votes
1answer
77 views

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 ...
1
vote
4answers
145 views

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 ...
4
votes
2answers
69 views

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 ...
1
vote
1answer
33 views

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$

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 ...
5
votes
2answers
96 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 ...
3
votes
2answers
54 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 ...
2
votes
1answer
32 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 ...
7
votes
2answers
77 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 ...
2
votes
1answer
63 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 ...
2
votes
2answers
111 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.
1
vote
3answers
148 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 = ...
8
votes
1answer
253 views

How is it possible that $\infty!=\sqrt{2\pi}$?

I read from here that: $$\infty!=\sqrt{2\pi}$$ How is this possible ? $$\infty!=1\times2\times3\times4\times5\times\ldots$$ But \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ ...
14
votes
10answers
1k 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.
1
vote
3answers
110 views

Product representations of the factorial function?

Is this the only product representation of the factorial function? $$ {n!} =\prod_{k=1}^{n} k $$
2
votes
2answers
553 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??
9
votes
2answers
175 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 ...
1
vote
0answers
57 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 ...
1
vote
0answers
59 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 ...
2
votes
2answers
153 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 ...
1
vote
1answer
130 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 ...
2
votes
2answers
230 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}}) - ...
3
votes
1answer
159 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} ...
8
votes
0answers
375 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 ...
2
votes
3answers
58 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 ...
3
votes
2answers
149 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 ...
12
votes
1answer
136 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 ...
2
votes
3answers
248 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! = ...
7
votes
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 $100!$ ($5$ ...
10
votes
2answers
244 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 ...
2
votes
1answer
196 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 $n\geq4$.
2
votes
3answers
222 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 ...
0
votes
1answer
116 views

Two questions on finding trailing digits in (large) numbers and one on divisibility

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 ...
2
votes
0answers
140 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
3
votes
0answers
127 views

Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

How can I prove the following, where $p$ is a prime and $x$ a positive integer? $$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$ I'm not sure if it is actually true, but I tested ...
5
votes
2answers
308 views

Find the remainder when $ 12!^{14!} +1 $ is divided by $13$

Find the remainder when $ 12!^{14!} +1 $ is divided by $13$ I faced this problem in one of my recent exam. It is reminiscent of Wilson's theorem. So, I was convinced that $12! \equiv -1 \pmod {13} $ ...
2
votes
1answer
175 views

If a function is a product of factorials of polynomials, how can I prove that this representation is unique?

If a function is a product of factorials of polynomials, how can I prove that this representation is unique? Specifically, let $F(x) = \prod_i{P_i(x)!^{a_i}}$, $F:\mathbb{N} \rightarrow \mathbb{Q}$, ...
13
votes
4answers
2k views

Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce ...
1
vote
1answer
148 views

Can someone explain to me the relationship between primorials and factorials and how that relation can be used to compute large factorials?

What I am trying to figure out is a way to compute large factorials, !1000000. For what it's worth luschny's computer algorithms do a very good job of it.
8
votes
1answer
4k views

Last non Zero digit of a Factorial

I ran into a cool trick for last non zero digit of a factorial. This is actually a recurrent relation which states that: If $D(N)$ denotes the last non zero digit of factorial, then ...
3
votes
1answer
354 views

Smallest positive integer

Let S(m) is the sum of the factorials of the digits of integer m. I try to find the smallest positive integer n with S(n)=111. My answer is 12334444. Is it right?
6
votes
2answers
901 views

On the factorial equations $A! B! =C!$ and $A!B!C! = D!$

I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this ...
5
votes
2answers
2k views

Derive a formula to find the number of trailing zeroes in $n!$ [duplicate]

Possible Duplicate: How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? I know that I have to find the number of factors of $5$'s, $25$'s, ...
7
votes
2answers
262 views

Finding all the numbers that fit $x! + y! = z!$

I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it. ...
6
votes
2answers
190 views

Closed form for $(p-n)!\pmod{p}$ where $p$ is prime

Does $(p-n)!\pmod{p}$ have a closed form for any $n>2$ when $p$ is prime? $(p-0)!=0 \pmod{p}$ $(p-1)!=-1\pmod{p}$ $(p-2)!=1\pmod{p}$