All Questions
Tagged with prime-factorization factorial
55
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Find sum of factorials divisible by the largest possible prime squared
Let $n$ be a positive integer.
Consider the following maximization problem :
Use each of the factorials $1,2,3!,\cdots ,n!$ at most once such that the resulting sum is divisible by $p^2$ , where $p$ ...
- 17.4k
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Prime factor wanted of the huge number $\sum_{j=1}^{10} j!^{j!}$
What is the smallest prime factor of $$\sum_{j=1}^{10} j!^{j!}$$ ?
Trial :
This number has $23\ 804\ 069$ digits , so if it were prime it would be a record prime. I do not think however that this ...
- 4,401
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Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.
The diagram shows Pascal's triangle down to row $10$.
I noticed that the product of the blue numbers is divisible by the product of the orange numbers. That is (including the bottom centre number $...
- 3,486
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Numbers $n$ such that $n!$ has all of its exponents odd
(First paragraph for motivation, second paragraph the actual problem)
I usually have to be on a stand as an embassador for my major (math) for highschoolers, and we usually have problems written on a ...
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Find smallest positive integer $n$ such that $n!$ is divisible by $p^k$ ($p =$ positive prime, $k =$ positive integer)
I have to find smallest positive integer $n$ in such way that $n!$ is divisible by $p^k$ ($p$ is always positive prime and $k$ is always positive integer).
$p$ and $k$ are given, $n$ is (obviously ...
- 605
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Finite many primes for every positive integer $b$?
Consider the function $$f(a,b):=\sum_{j=0}^a (bj)!=1+b!+(2b)!+\cdots +(ab)!$$
Given a positive integer $b$ , are there always only a finite number of positive integers $a$ such that $f(a,b)$ is prime ...
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Let a rational number $\frac{a}{b}$ in its lowest form where $a$,$b$ are integers, with $0 < \frac{a}{b}< 1$, b > 1. How many of these have $ab = 15!$
Consider a rational number $\frac{a}{b}$ in its lowest form where $a$, $b$ are integers, with $0 < \frac{a}{b}< 1$, b > 1. How many of these have $ab = 15!$
Solution Given in Book:
$15!=2^{11}...
- 18.9k
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Another Doubt Regarding Brocard's Problem, Specifically about Where I Can Proceed after Prime Factorisation
I know this can draw downvotes as it's not a complete solution or sounds more like an opinion poll (or perhaps this must have appeared elsewhere, in which case you're free to let me know), but I would ...
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If n is a positive integer that is four digits long and is relatively prime to 100!, why must n be prime?
Suppose there is some positive integer n that is four digits long and is relatively prime to 100! (meaning n and 100! have no common factors other than 1). n must be prime, but why?
100! is a ...
- 7,080
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Median factor of factorials
What is the order of $m(N)$, the median prime in the prime factorization of $N!$, as $N\to\infty$? For example, $m(6)=2$ because $6!=2^4\times3^2\times5$ and the median of $\{2,2,2,2,3,3,5\}$ is $2$.
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- 82.2k
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Factorials and Place Value
I recently came across this question from a non-calculator exercise. The units and tens place value digits I can see as $0$ and $0$ since in $12!$ we have $10*5*2=100$ but is there a way to find $a$ ...
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Find the power of a prime in the prime factorization of a large factorial
I have been trying to work through the following exercise:
Find the power of $5$ in the prime factorization of $2020!$.
So far I have worked out that the prime factorization of $2020$ is $2^2 \cdot 5^...
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Proof that each prime power of ${2n \choose n}$ is $\leq \log_p 2n$
I'm trying to work through this proof of the prime number theorem.
Def: Let $P_p(x)$ be the prime power of $p$ in the prime factorization of $x$. I.e. for any natural number $x$, $x = \prod_{p\in \...
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Total number of divisors of factorial of a number
I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where $a,...
CommunityBot
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perfect factors from the prime factorization of a large number
This is probably an easy question, but I don't know how to do it:
In the prime factorization of $30!$, how many perfect factors occur?
This is from a timed competition, any answers that take more than ...
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