All Questions
Tagged with prime-factorization divisibility
130
questions
9
votes
4
answers
1k
views
Determining Whether the Number $11111$ is Prime. Used Divisibility Tests.
I am asked to determine whether the number $11111$ is prime. Upon using the divisibility tests for the numbers 1 to 11, I couldn't find anything that divides it, so I assumed that it is prime. However,...
- 4,322
-1
votes
2
answers
180
views
fundamental theorem of arithmetic word problem [duplicate]
Hi here is the question I have in hand:
There are $1000$ empty baskets lined up in a row. A monkey walks by, and puts a banana in each basket, because this is a word problem,
and that is what a ...
0
votes
1
answer
99
views
Working with divisor function
So by Fundamental Arithmetic Theorem, any integer has a unique prime factorization into primes, written as:
$$n=p_1^{k_1}p_2^{k_2}p_3^{k_3}...p_r^{k_r}$$
From exponents $k_1,...k_r$ it is possible to ...
0
votes
1
answer
195
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Finishing the task to find the solutions of $\frac{1}{x}-\frac{1}{y}=\frac{1}{\varphi(xy)},$ where $\varphi(n)$ denotes the Euler's totient function
In this post I evoke a variant of the equations showed in section D28 A reciprocal diophantine equation from [1], using particular values of the Euler's totient function $\varphi(n)$. I ask it from a ...
user243301
10
votes
4
answers
13k
views
How to get all the factors of a number using its prime factorization?
For example, I have the number $420$. This can be broken down into its prime factorization of $$2^2 \times3^1\times5^1\times7^1 = 420 $$
Using $$\prod_{i=1}^r (a_r + 1)$$ where $a$ is the magnitude ...
- 453
6
votes
1
answer
237
views
On a product involving Ramanujan primes
We denote the $k$th Ramanujan prime as $\mathcal{R}_k$, that is the sequence A104272 from the OEIS as you can read from this Wikipedia. Then I was inspired in Richard K. Guy, Unsolved Problems in ...
user243301
1
vote
1
answer
58
views
Are there exactly $d$ distinct remainders when $x^{\frac{p-1}{d}}$ is divided by $p$?
Q1: Is it true that if $d$, if $d|p-1$, where $p$ a prime, then there for all $(x,p) = 1$, are exactly $d$ distinct remainders when $x^{\frac{p-1}{d}}$ is divided by $p$?
Q2: For what composite ...
- 20.8k
0
votes
1
answer
151
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Prove that $100|11^{10} - 1$ [duplicate]
I want to prove divisibility using factoring, So i need to show that $11^{10}-1$ can be written as prime factors of 100.
This is what I've tried:
$$11^{10}-1 $$
$$ (11^{5})^{2}-1$$
$$ (11^{5}-1)(...
- 91
1
vote
1
answer
2k
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Smallest number with at least $n$ divisors.
I have seen lots of posts on "exactly $n$ divisors" and understood the process as well, but I can't seem to find or come up with an algorithm, apart from brute force, for "at least $n$ divisors".
...
- 51
1
vote
3
answers
69
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Arithmetics: Number of divisors of an positive integer, their sum and product
So I've just started studying arithmetics a month ago, and I've came across this problem.
Let's say
$$
k = \prod_{m = 1}^n p_m^{i_m} \in \mathbb{N}
$$
and let $d(k)$ denote the number of $k$'s ...
- 81
2
votes
1
answer
142
views
The Chinese hypothesis revisited
In the past I tried to get different variations of the so-called Chinese hypothesis, see this Wikipedia (a disproven conjecture).
Today I wanted to combine in an artificious way also Wilson-Lagrange ...
user243301
0
votes
2
answers
66
views
Fractions of the form $\frac{a}{k}\cdot\frac{b}{k}\cdot\frac{c}{k}\cdots=\frac{n}{k}$
For any $n,k\in\mathbb{Z}^+$, $n\gt k$. I found that the majority of the time, either there are no positive integers, $a,b,c\dots,$ such that
$$\frac{a}{k}\cdot\frac{b}{k}\cdot\frac{c}{k}\cdots=\...
- 3,557
2
votes
3
answers
649
views
For which natural numbers are $\phi(n)=2$?
I found this exercise in Beachy and Blair: Abstract algebra:
Find all natural numbers $n$ such that $\varphi(n)=2$, where $\varphi(n)$ means the totient function.
My try:
$\varphi(n)=2$ if $n=3,4,...
- 4,794
0
votes
0
answers
708
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If $a\mid b$ then $\phi(a)\mid \phi(b)$ for $a,b\in\mathbb{N}$ [duplicate]
Hey I would like to show that
$a\mid b\Rightarrow \varphi(a)\mid\varphi(b)\qquad a,b\in\mathbb{N}$
where $\varphi(n)$ is the the totient function.
My try:
Let $a,b\in\mathbb{N}$ and $a\mid b$. ...
- 4,794
1
vote
0
answers
48
views
When $f(x)$ divides $d$ $f(x)=d(c+2ax+dx^2)\mod{N}$
Given $f(0)$ divides $d$ and $f(1)$ not, how to find other $x$ values that make $f(x)$ divisible by $d$?
$$f(x)=d(c+2ax+dx^2)\mod{N}$$
$a,c,d,x,N$ are positive integers
$c$ is a small number
$d$ is ...
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