Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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system of modular equations.

$x\equiv 2\pmod3$ $x\equiv 3\pmod 5$ $x\equiv 7 \pmod{11}$ How can I solve this system for $x$? I've tried all kinds of things using divisibility but no success. Any hints of solutions are greatly ...
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2answers
80 views

$\pi(x)$ Proof Clarification

In a proof from a number theory book that $${\pi(x) \over x}\le {2k \over x} + {\phi(k) \over k}$$ Where $x=kl+r$ with $0 \le r\lt k $ It is stated that $$\pi(x) \le k+(l-1)\phi(k) + r \le 2k+{x\over ...
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3answers
49 views

Smallest divisible repunits

A repunit of length k is a number containing k ones (1, 11, 111...). R(k) is defined to be the repunit of length k. A(n) is the least value of k such that R(k) is divisble by n (assuming gcd(n, 10) ...
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2answers
64 views

Forming natural numbers with positive consecutive integers

I'm trying to prove that any natural number N can be formed by adding at least two positive consecutive integers except for powers of 2. For example, using $\,N = 3$, $N = 1 + 2$. When experimenting ...
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3answers
270 views

prove that $\dfrac{\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number

Prove that $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that.
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0answers
16 views

Show that the equation has a natural solution [duplicate]

let $n$ be a natural number and $r$ , $s$ be rational such that $n=s^2+r^2$ show that there are natural numbers a,b such that $n=a^2+b^2$
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1answer
20 views

On $\gcd(a,x) = \gcd(b,x)=k \implies gcd(ab,x) = k$

Originally, I was examining $\gcd(a,x) = 1, \gcd(b,x) = 1$ and conjectured $\gcd(ab,x) = 1$. I think this is true, because I thought: Let $x = p_1^{a_1}\cdot p_2^{a_2}\cdot p_3^{a_3}\dots$ $a\neq ...
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6answers
255 views

Calculating remainder of $666^{666}$ when divided by $1000$.

I want to calculate the remainder of $666^{666}$ when divided by $1000$. But for the usual methods I use the divisor is very big. Furthermore $1000$ is not a prime, $666$ is a zero divisor in ...
12
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3answers
595 views

Perfect powers of successive naturals: Can you always reach a constant difference?

I was thinking about what happens if you take a sequence of consecutive squares, for example 1,4,9, 16. Taking the differences gives you another sequence, 7,5,3. And taking the differences between ...
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1answer
89 views

Prove that $2AB$ is square [duplicate]

Let $$A= 1! \cdot 2! \cdot 3! \cdots 1002!$$ $$B= 1004!\cdot 1005! \cdots 2006!$$ Prove that $2AB$ is square. Help guys, I tried, I really did but I couldn't.
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2answers
51 views

Square numbers in the form $1+4y$

I want to solve the equation $y+x=x^2$: $$ x^2-x-y=0 \\ x_{1;2}=\frac{1\pm \sqrt{1+4y}}{2} $$ However I want the solutions to be only natural numbers; the question then turns to find values of $y$ ...
2
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2answers
56 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
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2answers
63 views

Prove that to any three numbers positive integers [closed]

Prove that for any three positive integers, following equality holds $$\operatorname{lcm}(ab , bc , ca ) \cdot \gcd(a , b, c )=abc$$
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3answers
877 views

Students in a class, girls sitting with boys and boys sitting with girls

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
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0answers
189 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
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3answers
68 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [closed]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
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1answer
39 views

classification of groups of order $4p, p\ge 5$, need help finding automorphism

So I've been working on this problem for my qual prep class, and I have it all down except for one detail. I'm doing it by semidirect products, and with the Sylow $p$ group normal, choosing the ...
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1answer
56 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
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0answers
78 views

Rewriting $\#\{ b (\text{mod } 2n) | b^2 = D (\text{mod 4n}) \}$

Let $R^*(n) = \#\{ b (\text{mod } 2n) | b^2 = D (\text{mod 4n}) \}$ and $n = 2^{r_0} p_1^{r_1} \ldots p_s^{r_s}$ with $p_i$ prime and odd. Then we can rewrite $R^*(n)$ as $R^*(n) = R^*(2^{r_0}) \cdot ...
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4answers
92 views

Solving $x^2=17\pmod{128}$

I'm attempring to solve a congruence $x^2 \equiv 17\pmod{128}$ but not quite sure how to go about it. I see that $128 = 2^7$, but the Chinese Remainder Theorem doesn't apply to $\gcd > 1$. I found ...
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5answers
64 views

Is it allowed to define a number system where a number has more than 1 representation?

I was just curious; is it allowed for a number system to allow more than one representation for a number? For example, if I define a number system as follows: 1st digit (from right) is worth 1. 2nd ...
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0answers
44 views

sextic reciprocity and divisibility question

Regarding the question if $p|(2^{2(p-1)/6}+2^{(p-1)/6}+1) $ where $p$ is a prime of the form $7\mod 8 $ That is how far I got: $2^{(p-1)/6} \mod\ p\equiv x $ if the solution of $x^6\ mod\ ...
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2answers
68 views

Solve $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$

For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes?
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1answer
56 views

Is there any algorithm or something to solve $\phi\left(x\right)=n$ [closed]

solve for x, $\phi\left(x\right)=12$ where $\phi$ is euler's totient function($\phi\left(n\right)$ is the number of numbers less than n satisfying $hcf\left(n,i\right)$ with $1\leq i<n$. I'm ...
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0answers
65 views

Verification of Basic Proof in Spivak Calculus (Induction)

I have began working through Spivak's Calculus book and trying to do the problems at the end of the chapters. I am rather new to proof, so forgive the naivety of this type of question. I am wanting ...
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2answers
561 views

What is an “interesting integer” and are there uninteresting integers?

On a site, someone asked which number is most interesting and I answered, "Every number is interesting. Give me a number and I shall tell you why it is!". Now not going into philosophical ...
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3answers
62 views

How to calculate this expression and get an integer number?

Hello there I don't have idea how to calculate this: $$\left[\frac {116690151}{427863887} \times \left(3+\frac 23\right)\right]^{-2} - \left[\frac{427863887}{116690151} \times \left(1-\frac ...
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1answer
98 views

how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime?

Given that $$U_n=\underbrace{1\cdots1}_{n\text{ times}}$$ and $n >2$, how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime? Because $U_m= ...
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7answers
131 views

What is the last digit of $2003^{2003}$?

What is the last digit of this number? $$2003^{2003}$$ Thanks in advance. I don't have any kind of idea how to solve this. Except that $3^3$ is $27$.
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3answers
43 views

Problems on Pythagorean triangle

Show that there is one (no) Pythagorean triangle whose sides are in arithmetic (geometric) progression. The problem has two parts. There is one Pythagorean triangle whose sides are in arithmetic ...
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2answers
46 views

Show that $a^{16}-b^{16}$ is divisible by $133$ if $a$ and $b$ are both prime to $85$

Show that $a^{16}-b^{16}$ is divisible by $133$ if $a$ and $b$ are both prime to $85$ Since $(85, a)=1(17,5)$ and $(85, b)=(17,5)$ then $a^{16}-1\equiv (mod ~17)$, $a^{4}-1\equiv (mod~ 5)\implies ...
3
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2answers
71 views

Find the last two digits of $33^{100}$

Find the last two digits of $33^{100}$ By Euler's theorem, since $\gcd(33, 100)=1$, then $33^{\phi(100)}\equiv 1 \pmod{100}$. But $\phi(100)=\phi(5^2\times2^2)=40.$ So $33^{40}\equiv 1 ...
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4answers
76 views

Find remainder when $777^{777}$ is divided by $16$

Find remainder when $777^{777}$ is divided by $16$. $777=48\times 16+9$. Then $777\equiv 9 \pmod{16}$. Also by Fermat's theorem, $777^{16-1}\equiv 1 \pmod{16}$ i.e $777^{15}\equiv 1 \pmod{16}$. ...
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1answer
63 views

Find the missing digit in the number 23104*791

Find the missing digit in the number $23104*791$ if (i) it is divisible by $11$, (ii) it is divisible by $13$, (iii) it is divisible by $63$. (i) $23104*791=231 ...
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1answer
25 views

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$ I know that $a\equiv b (mod ~ m)$, $c\equiv d (mod ~m)$ implies $ac\equiv bd (mod ~m)$ but how ...
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2answers
62 views

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$. 71 is prime then Wilson theorem says that $(71-1)!+1=0 \mod ~ 71$ i.e $70!+1\equiv 0 \mod ~ 71$ then how to proceed further?
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7answers
99 views

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$?

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$ ? $7 \equiv 3 \pmod 4$ $7^2 \equiv 9 \pmod 4\equiv 1 \pmod 4$ $(7^2)^{16} \equiv 1^{16} \pmod 4$ i.e $7^{32} ...
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1answer
27 views

Find all values of $p$ such that $ax^2+bx+c \equiv 0 (\bmod p)$ have solution

Is there a general way to find all values of $p$ such that the congruence $ax^2+bx+c \equiv 0 (\bmod p)$ have solution, we can assume that $ax^2+bx+c =0 $ have solution.
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1answer
52 views

If p is an odd prime, show that $p^2 \equiv 1 \pmod 8 $

If p is an odd prime, show that $p^2 \equiv 1 \pmod 8 $. I know that odd numbers are of the form $2k \pm 1$. Then $p^2=(2k \pm 1)^2= 4k^2 \pm 4k +1$. But it does not help to solve.
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1answer
40 views

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$.

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$. Attempt: The pairs $m=1, ~ n=2$; $m=3, ~n=4$ satisfy the problem. But I need a ...
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2answers
50 views

Is this problem correct? [duplicate]

I have found another problem in my book. I have to prove that $$2^{70}+3^{70}$$ is divisible by 13. But I have proven that $2^{70}\equiv 12 (mod 13)$ and $3^{70}\equiv 3 (mod 13)$ so it is ...
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2answers
69 views

How can I find the remainder?

How can I find the remainder when $$(12371^{56}+34)^{28}$$ is divided by $111$. I have tried congruences modulo $111$ but without any success.
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6answers
729 views

Elementary number theory - prerequisites

Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really ...
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1answer
134 views

Can this expression be made true ? 2 _ _ _ _ = 2015

Make this expression true: 2 _ _ _ _= 2015 The underscores must be replaced by any 2 of of the operational symbols +, - , x, / (divide). And any 2 of the digits 0,1,2..9. So, you basically need 2 ...
0
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1answer
71 views

About the vertices of a regular polygon in the plane having rational coordinates [closed]

I have to prove that, except in the case $n=4$, the vertices of a regular $n$-agon in the Euclidean plane cannot have all rational coordinates $(x,y)$. Some idea?
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3answers
62 views

Determining the relative size of $a^n$ and $b^m$ without using logarithms

Example, which is larger $ 17^{105} $ vs $ 31^{84} $? Make the deternimination without resorting to logs, or Excel either.
1
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1answer
43 views

$p$ and $r$ are primes greater than $2$. $p+r$ vs $p+2r$, which could be a prime number?

For $p+2r$, a example would be $3$ and $5$. Since $6+5 = 11$, I am led to believe $p+2r$ to be the right answer. But I don't know how it works?
1
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1answer
65 views

Addition of points on elliptic curves over a finite field

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$: $$P + Q ...
0
votes
1answer
31 views

Factor RSA number $n$.

An RSA number $n=p\cdot q$, where $q=2\cdot d +1$, $d$ an odd integer, is given. Assuming $a \in \mathbb{Z}_n$ with $a^4=1$ and $a^2 \neq 1$. How can this information lead to finding $p$ and $q$? I ...
4
votes
2answers
65 views

Irrational numbers, induction

I have $\sqrt[3]{2}^{2^n}$. Can I prove that this number is irrational by showing that $3$ does not divide $2^n$?