-7
votes
0answers
39 views

Topic = Numbers ,{Simple But difficult for me :) } [on hold]

Question= There are how many different "a" natural number ?
3
votes
2answers
432 views

The sum of two irrational square roots

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ ...
-1
votes
0answers
40 views

How does this method work? [closed]

Let $n=16$ for an example: step 1: get set of prims from $1$ to $\sqrt{2n}: \{2, 3, 5\}$, step 2: get set of $n \mod 2, n \mod 3, n \mod 5: \{0, 1, 1\}$, setp 3: from $0$ to $n-3$, ...
0
votes
2answers
40 views

Need help finding a number x so that $\phi > 9x/10$?

I need help finding a number $x$ so that $φ(x) > 9x/10$? ($φ$ being Euler’s phi function.) I also need to find a number $x$ so that $φ(x) < x/3$?
3
votes
7answers
57 views

Suppose that $m \ge 0$ show that $49 \mid 5\cdot3^{4m + 2} + 53\cdot2^{5m}$

I've re-written the equation in a few different ways hoping for a few different approaches: $$49y = 5 \cdot 3^{4m + 2} + 53 \cdot 2^{5m} $$ I think the first equation has more potential, since it ...
1
vote
4answers
111 views

$7^n-1$ is divisible by $6$ for all natural number $n$ [closed]

How to prove $7^n-1$ is divisible by $6$ for all natural number $n$. Thanks for your help.
0
votes
1answer
186 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
0
votes
3answers
401 views

Finding the size of a list given its mean, and the mean when one number is added to the list

The mean of a list of $n$ numbers is $6$. When the number $17$ is added to the list, the mean becomes $7$. What is the value of $n$?
1
vote
2answers
68 views

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$.

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$. So, my thought process was that I could show that $2^{n} > n$ using induction, but ...
0
votes
2answers
33 views

calculate reverse number with 2 conditions

I can't find the reversed number of $2 \mod 13$ ($2^{-1}=?$) that is also a solution to $$5x = 2 \mod 13.$$ How can I find it? Thanks!
4
votes
6answers
163 views

Prove that in each year, the 13th day of some month occurs on a Friday [duplicate]

Prove that in each year, the 13th day of some month occurs on a Friday. No clue... please help!
0
votes
3answers
90 views

Inductive proof that $n^3+2n$ is divisible by $3$ for every integer $n$ [closed]

I have to prove this property with an inductive explanation If $n \ge 1$ and is an integer, then $3$ divides $n^3+2n$
1
vote
3answers
79 views

Find the remainder of $\frac{1! +2!+\, \dots\, +95! }{15}$.

I think I found my answer but I am looking for better ones
2
votes
0answers
81 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
1
vote
1answer
59 views

Proof by contradiction: logarithm

I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
0
votes
2answers
38 views

Dirichlet Characters and Chineese remainder theorem

Let $k=k_1 k_2$ s.t. $(k_1,k_2)=1$ and let $\chi$ be a dirichlet character mod $k$. I'm trying to prove that there exsists $\chi_1,\chi_2$ dirichlet characters mod $k_1,k_2$ respectively, s.t. ...
2
votes
8answers
169 views

How to solve this congruence $17x \equiv 1 \pmod{23}$?

Given $17x \equiv 1 \pmod{23}$ How to solve this linear congruence? All hints are welcome. edit: I know the euclidean Algorithm and know how to solve the equation 17m+23n=1 but I dont now how to ...
2
votes
5answers
67 views

Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
4
votes
2answers
79 views

For every integer, some multiple of it is of the form $99 \ldots 900 \ldots 00$

The goal is to prove that for every positive integer $ z$ there exists a positive integer $a$ such that $az = 99 \ldots 9900 \ldots 00$. Let $a = \frac {99 \ldots 9900 \ldots 00}{z}$ That ...
2
votes
3answers
156 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
4
votes
2answers
75 views

Will someone help with this number theory question?

Prove that if $2^{p}-1$ is prime then $$n=2^{p-1}(2^p-1)$$is a perfect number here is what i did: We need to prove the $\sigma(n)=n$ so $\sigma(n)=\sigma(2^{p-1})\sigma(2^p-1)$ since $2^{p}-1$ is a ...
6
votes
3answers
129 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
2
votes
0answers
32 views

Help with proof about solvability of linear diophantine equation.

Prove: If $a_1,...,a_n,c \in \mathbb Z,n \in \mathbb N$ then the equation $$a_1x_1+a_2x_2+\dots+a_nx_n=c$$ has integer solutions iff $\gcd(a_1,...,a_n)|c$. Proof: ($k=1,...,n$) $\Longrightarrow$ ...
1
vote
3answers
93 views

What is the highest $n$ such that $15^n\mid 100!\;?$

Can anybody please solve this problem? It's really confusing.
0
votes
1answer
67 views

Divisors of $n$

Please hint me. I have a question in my homework. let $n\in\mathbf{N}$ and $a$ and $b$ are two nontrivial divisors of $n$, so that $a\nmid b$ and $b\nmid a$ and $a>b$. I want to prove that there ...
4
votes
4answers
180 views

Prove that $(4/5)^{\frac{4}{5}}$ is irrational

Prove that $(4/5)^{\frac{4}{5}}$ is irrational. My proof so far: Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational. Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are ...
4
votes
6answers
177 views

prove that $2^{15} - 2^3 $ divides $ a^{15} - a^3$

Prove that $$2^{15} - 2^3 $$ divides $$ a^{15} - a^3$$ for any integer $a$. Hint: $$ 2^{15} - 2^3 = 5\cdot7\cdot8\cdot9\cdot13$$
0
votes
3answers
73 views

Find primes $p_1,p_2,..,p_6$ such that $1+\prod_{i=1}^{6}p_i $is not prime

Show that if$$ p_1, p_2, p_3, p_4, p_5, p_6 $$are primes, then $$1+\prod_{i=1}^{6}p_i$$ is not necessarily prime by using a specic example.
1
vote
1answer
71 views

Show that the converse of Fermats Little Theorem is false using a counter example.

Show that the converse of Fermat's little theorem is false using a counter example. Show that $$a^{561} \equiv a(mod p)$$ and hence that the converse of Fermat's little theorem is false???
4
votes
2answers
104 views

Question on gcd, is this true?

Let $a,b \in \mathbb{Z^+},\ a<b,\ d=\gcd(a,b)\ $ and $\ 1<d<a,\ x=\frac ad,\ y=\frac bd,\ x,y \in \mathbb{Z^+}.$ Suppose $a=a_1+a_2,\ b=b_1+b_2,\ a_1<b_1,\ d_1=\gcd(a_1,b_1)$ and ...
1
vote
2answers
29 views

Number of primitive roots $\pmod{m}$

I'm trying to find the number of primitive roots $\text{mod} 1300$ I thought this was calculated using $\phi (\phi (m))$ but I get that there are $128$ primitive roots, where as the solutions say ...
0
votes
1answer
32 views

Number Theory Divisibility Question

(From Math Challenge II Number Theory packet) Given that $a,b,n$ are positive integers. Assume that for any positive integer $k\neq b, (k-b)\mid(k^n-a)$, the which of the following must be true? ...
1
vote
0answers
42 views

Find primitive root mod 17

I have to list the quadratic residues of $17$ and find a primitive root. I have calculated that: Quadratic residues $mod(17)$ are $1,2,4,8,9,13,15,16$ How am I then meant to use this to obtain a ...
3
votes
3answers
69 views

Finding remainder on division by 2014

I'm trying to find the remainder when $6^{936}$ is divided by $2014$ I started thinking I could use Euler's theorem but then noticed that $6$ isn't prime, I then tried to split it into $6=2 \times 3$ ...
2
votes
5answers
83 views

How much zeros has the number $1000!$ at the end?

I know that it depends of the factors of five and two. But the number is too long to figure how much factos of five and two there are. Any hints?
2
votes
2answers
150 views

The number $25!$ has exactly 7 trailing zeros, true or false?

I don't know how to determine it... any hints?
1
vote
1answer
48 views

If $p=1\cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot … \cdot 2011$, then the units digit of $p$ is five

I know there is a $5$ on the sequence, but i don't know how and why his presence leads to the final units digit of the product.
0
votes
3answers
37 views

Find all positive integers $n$ such that $\log_2 (n)$ is a rational number.

For $\log_2(n)$ to be a rational number, I started by stating that: $\log_2(n)=\dfrac{a}{b}$ such that $a,b \in\mathbb Z$ and $b \neq 0$ but I really don't know what step to take next?
2
votes
3answers
117 views

Prove that $\sqrt{7}+\sqrt{3}$ is irrational [duplicate]

Is there a method by which we can prove that $$\sqrt{3}+\sqrt{7}$$ is irrational. It's obviously an irrational number, but I want to prove that mathematically.
0
votes
1answer
34 views

Is 2(2k-1) is a perfect square for positive integer k?

For positive integer $k$, let $M = 2(2k-1)$, which of the following must be true? (a) $M$ is not a perfect square for any $k$. (b) There are infinitely many $k$ such that $M$ is a perfect square. ...
-2
votes
2answers
51 views

Find the natural numbers $n$ in which $n^2$ divides $584$? [duplicate]

I'm trying to find the natural numbers $n$ in which $n^2$ divides $584$ ? i tried all the ways i know but i get stuck.
1
vote
2answers
87 views

Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
6
votes
2answers
258 views

Sum of Digits Question

If A is the sum of the digits of $5^{10000}$, B is the sum of the digits of A, and C is the sum of the digits of B, what is C? I know it has something to do with mod 9, but I'm not sure how do use it ...
0
votes
1answer
23 views

Let $a$ and $b$ be coprime positive integers. Prove that, for any integer $n$, there exist integers $s$ and $t$ such that $sa + tb = n$

I always sort of took this fact for (well..) fact. Can someone help me with the proof? Does this question have something to do with modulus? Since $a$ and $b$ are coprime ($gcd$ = 1), multiplying ...
0
votes
1answer
58 views

Number theory problem exercise? [closed]

Find all natural numbers $N$ so that $\varphi(N)=24$ where $\varphi$ is Euler's function.
2
votes
1answer
64 views

Help with $\sum_{d\mid n}τ(d)^2=\sum_{d \mid n}τ(d)^3$

I am doing some exercises on number theory on multiplicative number theoretic functions and I have some problems with the multiplication on sums like the sum $\sum_{d\mid n}(τ(d))^2$ where $d$ is a ...
3
votes
0answers
54 views

Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
2
votes
0answers
72 views

How to prove sum of two numbers of the two following forms can be equals to sum of two numbers not of the forms?

The two forms are: $\ 3x^2 + (6y-3)x - y\ $ $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+}$ For example: $\ \ \ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $,of the two forms $\ ...
1
vote
3answers
54 views

Diophantine equations problem/exercise 3

Find all the pythagorean tripples (x,y,z) with x=40. Well I started with the known formulas for the pythagorean tripples but got me nowhere. Or I was not able continue the thought process required. I ...
2
votes
1answer
34 views

Diophantine Equations problem 2

Find all the solutions to the Diophantine equation x^2+y^2=2(z^2) .I do not have alot of expirience on Diophantine equations and i do not know how to approximate them.I can see that the tripples of ...