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

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Find all natural numbers $a,b,c$ such that $abc+ab+c=a^3$

Find all positive integers $a,b,c$ such that $$abc+ab+c=a^3$$ My try: Clearly $c=ak$ $abk+b+k=a^2$ $b=\frac{a^2-k}{ak+1}$ is an integer but I am not getting anything further
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1answer
34 views

How does the Euler Totient Function apply here?

How many positive integers $< 2013$ are divisible by $2$ Can I somehow use Euler's Totient function to find this?
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1answer
9 views

On Inequality Concerning Deficient NUmbers

By Definition a positive integer $N$ is d-deficient if $\sigma(N)=2N-d$. Am I correct if I say that the inequality $N>d$ always hold for this definition? Here is my attempt to show that it is ...
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2answers
49 views

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$?

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$? This was an olympiad question. I thought of writing a number $x \le 2012$ in the form: $x = 2^{a}3^{b}4^{c}5^{d} = ...
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2answers
106 views

Is there an obvious reason why $4^n+n^4$ cannot be prime for $n\ge 2$? [duplicate]

I searched a prime of the form $4^n+n^4$ with $n\ge 2$ and did not find one with $n\le 12\ 000$. If $n$ is even, then $4^n+n^4$ is even, so it cannot be prime. If $n$ is odd and not divisible by ...
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1answer
35 views

For how many integers is this a prime number?

For how many integers $n$ is: $9 - (n-2)^2$ a prime number? I want to try this using a rigorous definition of prime number/ actual problem rather than try-error? Please only give hints, so I can do ...
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3answers
21 views

Proving if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$

Let $a,b,c\in \mathbb Z$. Prove that if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$. I get that sometimes this can acutally be false. Define ...
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3answers
56 views

Proving that if $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even.

Let $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even. My attempt: If one or two numbers of $a,b,c$ are even then we're done, so we'll have to show that at least one of them is even. ...
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2answers
26 views

Congruence definitions equivalence

We say that $x$ is congruent to $y$ modulo $z$ when $$x\equiv y\pmod z \iff x \pmod z = y \pmod z$$ Another definition is $$\quad x \equiv y \pmod z \iff \exists k \in \mathbb{Z}: x - y = k z$$ Why ...
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1answer
13 views

For the following number, state the base represented as t?

$1011 \textrm{(base }t) = 4931 \textrm{(base 10)}$ I have to find $t$, which is the base of 1011. I do the following: $4931 \textrm{(base 10)} = 4 \times 10^3 + 9 \times 10^2 + 3 \times 10^1 + 1 ...
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2answers
49 views

Suppose $\sqrt2=a/b$, with $gcd(a,b)=1$. Then $3|(a^2+b^2)$ implies that $3|a$ and $3|b$,

Suppose $\sqrt2=a/b$, with $\gcd(a,b)=1$. Then $a^2=2b^2$, so that $a^2+b^2=3b^2$. But $3|(a^2+b^2)$ implies that $3|a$ and $3|b$, a contradiction. I don't understand how $3|(a^2+b^2)$ implies that ...
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3answers
94 views

Show that $n^4+4$ is not a prime number

How do you show that for all $n ∈ N, n ≥ 2,$ $n^4 + 4$ is not a prime number? My attempt: I see that whatever number $n^4+4$ makes when $n$ is an even number would result to an even number. Thus ...
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1answer
25 views

I need Sophie Germain primes in the 7-digit range

About a year ago some one asked if there was a list of ALL Sophie Germain primes. One answer pointed the questioner to: vaxasoftware.com/doc_eduen/mat/primsophie_en.pdf. That list only goes up to ...
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0answers
27 views

Given a positive integer n show that there exists one and only one pair of integers h and k with 0 ≤ h < k such that n = 1/2 k(k − 1) + h.

Given a positive integer $n$ show that there exists one and only one pair of integers $h$ and $k$ with $0 \leq h < k$ such that $n = \frac{k(k-1)}{2}+h$. I don't really know how to approach this ...
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1answer
34 views

For what natural number $n$ is the following inequality true: $2^n \geq 2\cdot n^2$?

Can you solve this by using induction? The inequality is true for $n = 1$, but is false until $n = 7$. After the induction step I got $$2^n \geq n^2 + 2n + 1.$$ If you take the limit as $n$ ...
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2answers
23 views

Can someone help me prove that $\tau(n)$ is odd [duplicate]

Can someone help me prove that $\tau(n)$ is odd if and only if $n$ is a perfect square. So basically I have to prove that $\tau(n)$ is odd iff $n = k^2$ for some integer $k$.
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1answer
26 views

My proof that there are primitive roots modulo $p^2$

Let $p$ be a prime number. I'd like to prove that there are primitive roots modulo $p^2$. Could someone check this argument? Note that if $r\in\mathbb Z$ is a primitive root modulo $p^2$, it must ...
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4answers
52 views

Show that if $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$.

Show that if $m$ and $n$ are integers such that $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$. I am not sure where to begin.
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5answers
92 views

Prove that $2^{3^n} + 1$ is divisible by 9, for $n\ge1$

Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$. Work of OP: The thing is I have no idea, everything I tried ended up on nothing. Third party commentary: Standard ideas to attack ...
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2answers
19 views

Congruence with additional conditions. [on hold]

Let $$\left(ac \equiv bc \pmod m\right) \wedge \left(gcd(c,m) = d\right) \implies a \equiv b \pmod {\frac{m}{d}} $$ Is it true? Why? Thanks in advance.
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2answers
38 views

Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
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1answer
48 views

How many 10 digit numbers are there so the sum of the digits is $2$?

How many 10 digit numbers are there so the sum of the digits is $2$? $abcdefghij$ is the 10 digit number. By default, $a=1$ is a must. $= 1bcdefghij$ Now we need: $bcdefghij = 1$ How can I solve ...
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1answer
74 views

Using fermats last theorem in a proof

Question: If $x,y,z,n$ are natural numbers, $x,y,z,n>1$, with $x^n +y^n=z^n$ then show that $x,y,z$ are all greater than $n$ Here to prove this i would like to use Fermat's last theorem, to ...
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1answer
29 views

Sum of divisor powers?

A given number is divisible by 2, 3, and 5, and has altogether 2013 divisors. The smallest such number is $2^N \cdot 3^M \cdot 5^p$ where $N + M + P=$? I would $N + M + P = 2012$ because by a ...
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2answers
47 views

Smallest integer $x$ for which 10 divides $2^{2013} - x$

Find the smallest integer $x$ for which 10 divides $2^{2013} - x$ Obviously, $x \equiv 2^{2013} \pmod{10}$ But how can I reduce $x$?
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32 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
4
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1answer
84 views

If $\frac{a+1}{b}+\frac{b}{a}$ is an integer then it is $3$.

If $\frac{a+1}{b}+\frac{b}{a}$ is an integer for positive integers $a,b$ then prove that this integer is $3$. I reduced the to prove that if $\frac{c^2+d^2+1}{cd}$ is an integer then it is $3$ where ...
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2answers
31 views

$2^{n+1}|2^{2^n}$ and $2^{2^n}+1|2^{2^{n+1}}-1$

$2^{n+1}|2^{2^n}$ and $2^{2^n}+1|2^{2^{n+1}}-1$ I have not been able to show the above. I would greatly appreciate any help.
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1answer
15 views

Deduce that the number of divisions in the Euclidean algorithm is at most $2n + 1$

Theorem. If $a > 0$ and $b$ is arbitrary, there is exactly one pair of integers $q, r$ such that the conditions $b = qa + r, 0 \leqslant r < a$, hold. Repeated application of this theorem ...
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1answer
35 views

Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.

Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $\mid N \mid$ or $\frac{\mid N \mid}{4}$, respectively, is $1$ or prime. I have an idea of how to show ...
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1answer
37 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
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2answers
54 views

$\mathrm{lcm}(b,c)$ from $\mathrm{lcm}(a,b)$ and $\mathrm{lcm}(a,c)$

Given that lcm$(a,b)=60$ and lcm$(a,c)=270$, find lcm$(b,c)$ I believe you're supposed to use the rule lcm$(a,b)=p_1^{\text{max}(r_1,s_1)}\cdots p_m^{\text{max}(r_m,s_m)}$ Here's my work so far: ...
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1answer
9 views

“Multivariable” version of this lemma about showing analytically that a number is irrational.

Lemma: let $\alpha \in \mathbb{R}^+$ and $p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N}$ such that $\left|\alpha q_n - p_n \right| \neq 0$ for all $n \in \mathbb{N}$ and $$ \lim_{n \rightarrow ...
3
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1answer
58 views

Could someone take a crack at this number theory problem?

The question is stated as follows: If $\mathrm{gcd}(a,m)=1$ and $X$ is a complete residue system $\bmod m$, then the set obtained by multiplying each member of $X$ by $a$ is also a complete residue ...
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1answer
38 views

Numbers that can be represented by 32 bits

A typical computer 'word' is either 32 or 64 bits long. For each of the following encoding, determine the range of numbers (in base 10) that can be represented with (i) 32 bits and ...
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1answer
34 views

How to prove $V(5x^2+6xy+2y^2-2yz-z^2)$ is empty

Let $V/\mathbb{Q}$ be the projective variety $V:5x^2+6xy+2y^2=2yz+z^2$. I want to prove $V(\mathbb{Q})$ is empty. Given $[x,y,z]$ in $V$, WLOG assume $x,y,z\in \mathbb{Z}$ and $\gcd(x,y,z)=1$. ...
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2answers
48 views

Find the number which is the sum of different consecutive integers

Problem: Find $n$ such that $n>200$ $n$ can be written like the sum of of $5$, $6$, and $7$ consecutive integers I'm currently studying modular arithmetic so I tried to solve witusoinh it. ...
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1answer
54 views

Relationship between increasing integer sequences

Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal ...
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2answers
36 views

How to solve this quartic congruence?

Given $x^4 + 36x^3 - 19x^2 + 11x - 14 \equiv 0 \pmod{5}$. How would one go about solving such an congruence equation? Maybe it's possible to reduce this to a quadratic congruence? I can't figure it ...
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2answers
36 views

Let $t_n$ denote the $n$th triangular number. For what values of $n$ does $t_n$ divide $t_1^2+t_2^2+ \cdots +t_n^2$

Let $t_n$ denote the $n$th triangular number. For what values of $n$ does $t_n$ divide $t_1^2+t_2^2+ \cdots +t_n^2$. The hint says that because $t_1^2+t_2^2+ \cdots +t_n^2 = t_n(3n^3 + 12n^2 + 13n + ...
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2answers
35 views

Criteria for the existence of zero-divisors and idempotent elements in the integers modulo $n$

I need help in establishing or at least deciding the validity of the following two criteria: There are in the ring $Z_n$ non-trivial zero divisors if only if $n$ is divisible with some square. ...
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4answers
70 views

Proof: if p is prime, and 0<k<p then p divides ${p \choose k}$

Question : IF p is prime, and 0< k< p show that $ p | {p \choose k}$ ${p \choose k}$ can be rewritten as: $${p(p-1)(p-2)... (p-(k-1))(p-k)! \over (p-k)! k(k-1)(k-2)...3.2.1}$$ Now the (p-k)! ...
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4answers
69 views

Solve $3x \equiv 17 \pmod{2014}$

Solve $$3x \equiv 17 \pmod{2014}$$ So first I suppose $3^{-1} \pmod{2014}$ $2014 = 671(3) + 1 \implies 1 = 2014 - 671(3)$ But this gives $3^{-1} = 1 \pmod{2014}$ which is incorrect?
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3answers
28 views

Find the Inverse Modulus using Euclid's algorithm

I asked this before, but unfortunately, I didnt know the methods, nor was the questions phrased properly. Find the inverse of $4258 \pmod{147}$ Using Euclidean Extended Algorithm. Begin By Stating ...
4
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4answers
227 views

Find how many positive divisors a number has. What would you do?

Recently I learned an amazing thing. Suppose you are given a number and you have to find how many positive divisors it has. What would you do ? Solution: Suppose you select $12$. It has ...
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3answers
32 views

Solve diophantine equation using modular arithemtic

Solve for integers, $x, y$ $4258x+147y=369 \implies 4258x \equiv 369 \pmod{147}$ I got this question from SE, but I want to try this approach. I suppose we will find the inverse modulus of $4258 ...
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3answers
57 views

Last 2 digits of $2345^{369}$

http://i.stack.imgur.com/hte3J.jpg This webpage says last 2 digits of $2345^{369}$ is $75$. But considering only last 2 digits: $45^1 = 45$ $45^2 = 25$ $45^3 = 25$ The last 2 digits are always ...
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3answers
27 views

Can I conclude there's no $x/y$ such that $(x/y)^2=-1$ mod 3

Suppose $x^2+y^2=0$ mod 3. I want to show 3 divides $x$ and $y$. Assume $(y^2,3)=1$. Dividing $y^2$ gives $(x/y)^2=-1$ mod 3. Here I want to use the fact that $-1$ is not congruent to any square mod ...
0
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2answers
33 views

Show if $k$ is an integer, then $\sqrt[n]{k}$ is rational if and only if it is an integer.

$(i)$ Show that if the reduced fraction $a/b$ is a root of the equation $c_0x^n + c_1x^{n-1} + \cdots + c_n = 0, $ where $x \in \mathbb{R}$ and $c_0,\ldots,c_n \in \mathbb{Z}$ with $c_0 \ne 0$, ...
7
votes
3answers
356 views

Finding the number of divisors of a number?

How can I find the number of divisors of $2011\times2012\times2013\times2014+1$?