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

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1answer
146 views

Is 0.999999999 the same as 1? [duplicate]

0.99999999999 divided by 1 = 1 / 9 = 0.11111111111111 1 divided by 9 = 1 / 9 = 0.111111111111 So, does that make 1 = 0.99999999999999
1
vote
2answers
95 views

Help finding all the solutions of the diophantine equation: $ x^2+4(ab)^n=y^2$

I need help finding the solutions of the diophantine equation: $ x^2+4(ab)^n=y^2$. Please be aware the only one that I could find was $ x= \pm (a^n-b^n), y=\pm (a^n+b^n)$ using the quarter squares ...
3
votes
5answers
249 views

How to show that $(3k+2,5k+3)=1$ for all $k\in\mathbb{Z}$

I think I'm on the right track, but I can only figure out how to prove for a specific $k$ of my choosing... I don't know how to generalize it for all $k$: Assume $(3k+2,5k+3)=1$. Therefore, there ...
1
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0answers
45 views

Show by induction… Help

Let $a,n\in\mathbb{N}$, show that there exists $m\in\mathbb{N}$, such that $(a+1)^n=ma+1$ I tried to do by induction on $n$, but found it a bit strange the demonstration. ...
3
votes
2answers
642 views

Prove that the Diophantine equation $ax+by+cz=e$ has a solution if and only if $(a,b,c)\mid e$.

I have an intuitive idea about how this is going to work, but I don't know if I'm writing it properly using proper math language and theorems. I am most uncomfortable with the second half of the proof ...
7
votes
1answer
196 views

$n^3+7$ cannot be a square number.

Let $n$ be a positive integer. What I have to show is that $n^3+7 \neq k^2$ for any integer $k$. I assumed that $n^3+7=k^2$ for some integer $k$. What I did : $$(n+2)(n^2-2n+4)=k^2+1$$ $$k^2=-1 ...
3
votes
2answers
3k views

the nth root of n!?

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
2
votes
3answers
46 views

Why is this hypothesis necessary?

The following seems to be a common lemma for proving Fermat's little theorem: Let $1\leq k\leq p-1$ and $p$ be a prime. Then $\binom{p}{k}\equiv 0\pmod p$. We have ...
0
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4answers
148 views

Find all rational values of $x$, at which $ \sqrt{x^2 + x + 3}$ is rational

How to find all rational values of $x$, at which $y = \sqrt{x^2 + x + 3}$ is rational?
3
votes
1answer
412 views

How do I solve a linear Diophantine equation with three unknowns?

Find one integer solution to the Diophantine equation \begin{equation*} 18x+14y+63z=5. \end{equation*} If this were only a linear equation over $\mathbb{Z}^2$, then I could easily solve it by ...
6
votes
2answers
155 views

Are there identities which show that every odd square is the sum of three squares?

I am looking for algebraic identities of the form $$ (2n+1)^2 = f(n)^2 + g(n)^2 + h(n)^2, $$ where the functions are polynomials in $n$. EDIT: Evidently $(6k)^2 = 36k^2$ is trivially the sum of ...
2
votes
5answers
2k views

Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)

I am lost. So far... If $\gcd (a,b) = 1$, by Bezout's Formula $ax + by = 1$ If $c|(a+b)$, then $cf = a+b$ Then, $a (x-y) + cfy = 1$ $b(yx) + cfx = 1$ Am I on the right track? Any ...
1
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2answers
134 views

Distinct Mersenne numbers are coprime

How can you prove that if $p$ and $q$ are distinct primes, then the following holds?: $$(M_p,M_q)=1$$ Note: $M_n=2^n-1$, with $n$ prime number
0
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1answer
28 views

We suppose: $a, b, c \in \mathbb{N}$ and $b$ and $c$ are multiples of $a$ then $b+c$ is a multiple of $a$.

Are those statements true? We suppose: $ a, b, c \in \mathbb{N}$. We suppose: $b$ and $c$ are multiples of $a$ then $b+c$ is a multiple of $a$. We suppose: $b$ and $c$ are multiples of $a$ and $(b ...
2
votes
3answers
92 views

If $a$ and $b$ are odd integers, then $\sqrt{a^2+b^2}$ is irrational

If $a,b\in\mathbb{N}$ are odd then demonstrate: $$ {\sqrt{a^2 + b^2}} \not\in \mathbb{Q}$$ I try to guess that $$ {\sqrt{a^2 + b^2}} \in\mathbb{Q}.$$ Then i write $$ {\sqrt{a^2 + b^2}= m/n}.$$ ...
-1
votes
1answer
57 views

Finding out if an equation has a solution in $\mathbb{Z_3}$

Prove $x^n + y^n = z^n$ has solutions for $x, y, z \in \mathbb{Z_3}$ only for odd $n$. So for example, $1^2 + 2^2 = 5$ and $5$ doesn't have a square root, which means it doesn't have a solution. ...
0
votes
4answers
155 views

Why is $n^2 - 2$ never a multiple of $3$?

I know that for any $n$, $n^2 - 2$ is never a multiple of $3$. I feel like this is a rather simple proof, but I cannot figure out how to manipulate the definition of a multiple of $3$: $n$ is a ...
1
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0answers
121 views

$ p = a^{2} + ab +b^{2} \ a, b \in \mathbb{Z} $

Let $p \neq 3 $ be a prime. Prove that $ p = a^{2} + ab +b^{2} \ a, b \in \mathbb{Z} \iff p \equiv 1 \ mod \ 3$. The $\rightarrow $ direction is easy. For the other implication, I considered ...
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0answers
82 views

Existence of a Vampire number on the form $v = xy = a^bb^a$?

A number $v = xy$ with an even number $n$ of digits formed by multiplying a pair of $n/2$-digit numbers (where the digits are taken from the original number in any order) $x$ and $y$ together. ...
5
votes
2answers
1k views

Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$

So, if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$ means $1=ax+by$, and want to show $a^2x+b^2y=1$. By squaring $1=ax+by$ both sides, I get, $1=(ax)^2+b(2axby+by^2)$. It doesn't help my proof. Please help ...
3
votes
2answers
99 views

Generating Pythagorean Triples S.T. $b = a+1$

I am looking for a method to generate Pythagorean Triples $(a,b,c)$. There are many methods listed on Wikipedia but I have a unique constraint that I can't seem to integrate into any of the listed ...
2
votes
2answers
431 views

Proving that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$ [duplicate]

Let $a$, $b$, $m$ be integers. Prove that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$. I need help proving this. I am not sure what to do. Thank you for all of the edits. ...
6
votes
3answers
389 views

Is $(36^{36}+41^{41})/77$ an integer?

Let $$a=\frac{36^{36}+41^{41}}{77}.$$ Is $a$ an integer? I know that: The last digit ofr $41^{41}$ is $1$. The last digit of $36^{36}=6^{72}$ is $6$. How can I use this information to answer my ...
1
vote
1answer
144 views

Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?

A positive integer $N$ is said to be $k$-multiperfect if $$\sigma(N) = kN$$ where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer. (The case $k = 2$ reduces to the ...
1
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5answers
78 views

Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.

Form the assumption, we can say $b=ak$ ,$k$ integer, $a=bm$, $m$ integer. Intuitively, this conjecture makes sense. But I can't make further step.
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4answers
177 views

How can I find the formula used to produce this number?

In a game, each character has different attributes with values to them. The attributes are things like Strength and Speed and are graded on a scale of 1-100. The game uses a formula to produce an ...
8
votes
1answer
137 views

Integer ordered pairs (x,y) satisfying $x^2 - y! = N$

This question arose by looking at a similar question, which has $N = 2001, 2013$. In it, my solution was that since we have a prime $p$ (in this case 3) which divides $N$ but $p^2$ doesn't, hence we ...
1
vote
2answers
387 views

Remainder when $26^{3008} + 3008^{26}$ is divided by $4$

I want to find the Remainder when $26^{3008} + 3008^{26}$ is divided by $4$. What should I do? Even though I've included the tag modular arithmetic I've very limited knowledge about it. How should I ...
3
votes
3answers
1k views

Use mathematical induction to prove that 9 divides $n^3 + (n + 1)^3 + (n + 2)^3$; Looking for explanation, I already have the solution.

I have the solution for this but I get lost at the end, here's what I have so far. basis $n = 0$; $9 \mid 0^3 + (0 + 1)^3 + (0 + 2)^2 ?$ $9 \mid 1 + 8$ = true Induction: Assume $n^3 + (n + ...
2
votes
1answer
52 views

Ratio Proportion question

What must be subtracted from each term of the ratio $3:7$ so that the ratio becomes $2:5?$ My attempt: Let two numbers be $3x$ and $7x.$ So, $\frac{3x-y}{7x-y}=\frac25 \implies 15x-5y=14x-2y ...
1
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3answers
167 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 = ...
1
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0answers
52 views

How do I construct this multiplication table?

I was told to write out a multiplication table for $Z_3[\sqrt{3}]$. I'm not sure what that means... I'm used to just writing it out for $Z_3$
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0answers
120 views

is number theory a good place to start learning maths after a long break?

I haven't studied maths since my engineering degree some 15 years ago. At the time, I enjoyed maths and found it quite easy. However, now I feel that I have lost everything I learnt and I'm really ...
6
votes
3answers
453 views

$1^n +2^n + \cdots +(p-1)^n \mod p =$?

Calculate for every positive integer $n$ and for every prime $p$ the expression $$1^n +2^n + \cdots +(p-1)^n \mod p$$ I need your help for this. I don't know what to do, but I'll show you what I ...
1
vote
1answer
59 views

Quadratic Congruences Number Theory Question

Prove that every odd prime divisor of $n^2+100$ is of the form $12k+1$ or $12k+5$. I'm not sure how to do this.
2
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3answers
136 views

Demonstrate that if $ {\sqrt{a}} + {\sqrt{b}}\in\mathbb{Q}$ then $ {\sqrt{a}}\in\mathbb{Q}$ and ${\sqrt{b}}\in\mathbb{Q}$

$a,b\in\mathbb{Q}$ if $$ {\sqrt{a}} + {\sqrt{b}}\in\mathbb{Q}$$ demonstrates: $$ {\sqrt{a}}\in\mathbb{Q}$$ $${\sqrt{b}}\in\mathbb{Q}$$ I try to solve it with the property: If x, y ...
0
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0answers
28 views

Divisible polynomial and first term [duplicate]

How do I show that if $b^2+ab+1$ divides $a^2+ab+1$ for a, b are poaitive integers. Then $a=b$?
0
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1answer
55 views

Number theory in divisibility [closed]

Prove or disprove the following. Let $a_1,\ldots,a_r$ be positive even integers, and let $b_1,\ldots,b_s$ be positive integers. If $r\geq s+3$ and $a_i > b_j$ for all $i$ and $j$, then the quotient ...
1
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0answers
31 views

Are there any simple functions which map $\mathbb Z^n\to \mathbb Z\setminus \{k\}$ for given integer $k$?

Obviously, a function could be explicitly constructed as the set of all points in $\mathbb Z^n$ and what they are mapped to such that the given integer $k$ is not in the range. I am hoping to find a ...
2
votes
2answers
85 views

Number theory problem on numbers [closed]

Prove that if $n$ is divisible by $11$ and $n'$ is obtained from $n$ by inserting two identical digits between consecutive digits of $n$, then $n'$ is also divisible by $11$. For example, since $407$ ...
3
votes
2answers
92 views

What is the remainder of dividing $(116+17^{17})^{21}$ by $8$?

What is the remainder of dividing $(116+17^{17})^{21}$ by $8$? How to solve this? Solving the congruence (find the value of $a$) in $$(116+17^{17})^{21}\equiv a\pmod8$$ and how is this done?
2
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2answers
122 views

Number theory problem in induction

Without using the fundamental theorem of algebra (i.e. the prime factorization theorem), show directly that every positive integer is uniquely representable as the product of a non-negative power of ...
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3answers
151 views

What is the remainder of dividing $14^{256}$ by $17$?

What is the remainder of dividing $14^{256}$ by $17$? $$14^2\equiv 196\equiv 9 ...
5
votes
2answers
58 views

Image of an integer polynomial

How do I describe all integers that can be written in the form $$(x+y)^2+5x+3y$$ for some integers $x$ and $y$?
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2answers
102 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
1
vote
1answer
163 views

ZFC-universe without non-standard natural numbers?

I assume, that we all have (beside any set-thoretic background) a good intuition, what "the usual natural numbers" are, although we seem not to be able to describe them precisely. So having said ...
4
votes
1answer
110 views

Prove that $1^2 3^2 5^2 \cdots (p-4)^2 (p-2)^2 \equiv (-1)^{(p+1)/2} \pmod p$.

Let $p$ be an odd prime number. Prove that $$1^2 3^2 5^2 \cdots (p-4)^2 (p-2)^2 \equiv (-1)^{(p+1)/2} \pmod p.$$ I know I can use Wilson's Theorem somehow. It would make sense if I could show that all ...
1
vote
3answers
49 views

If $a,p\in\mathbb{N}$ with $p$ prime, have to show that if $a²\equiv1\pmod p $, then $a\equiv1\pmod p$ or $a\equiv p-1\pmod p$

If $a,p\in\mathbb{N}$ with $p$ prime, have to show that if $a²\equiv1\pmod p $, then $a\equiv1\pmod p$ or $a\equiv p-1\pmod p$ I'm studying congruence, and I have no idea where to start this ...
2
votes
1answer
230 views

Number theory GCD word problem

An oil company has a contract to deliver 100,000 litres of gasoline. Their tankers can carry 2,400 litres, and they can attach one trailer carrying 2,200 litres to each tanker. All the tankers and ...
0
votes
1answer
33 views

If $a\equiv b\pmod m$ and $c+d\equiv 0\pmod m$ then $ac+bd\equiv 0\pmod m$ [duplicate]

If $a\equiv b\pmod m$ and $c+d\equiv 0\pmod m$ then $ac+bd\equiv 0\pmod m$. The response, posted below is correct??