# Tagged Questions

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### Yet another product of irrational numbers

Let $~\alpha~$ and $~\beta~$ be irrational numbers such that $$~\alpha \notin \{\beta, -\beta\}$$ and $$~\alpha \notin \left\{\frac{1}{\beta}, -\frac{1}{\beta}\right\}$$ I suppose that in this case ...
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### Square grid , sum of elements

I am trying to solve the following problem : Find all the positive integers $n$ and $k$ such that it is possible to write integers in an $n \times n$ grid so that the sum of all elements in the grid ...
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### Particular number is divisible by 11

Let $\mathcal{N} \$ be a natural number of the form $\mathcal{N}=\textrm{dcba}$ ($a$ being the number of units $b$ the tens digit $c$ the hundreds digit and $d$ the thousands digit). On what ...
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### congruence issue

I need to understand why this : $$(1+4+\ldots+4^{n−1})\equiv n \pmod3$$ Is that because \begin{align} 1&\equiv -2 \pmod3\\ 4&\equiv 1 \pmod3\\ 4^{2}&\equiv1 \pmod3\\ ...
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### The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
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### Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
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### Find out the no of digits in product between some prime.

How many digits are there in? $2^{17}*3^{2}*5^{14}*7$. help me.
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### Given a set of nonnegative numbers, put $\pm$ between them to minimize the magnitude of the result

Let's say I have a finite set of non-negative numbers. I have to put $+$ or $-$ between the numbers, in order to minimize the absolute sum.(i.e the sum has to be closest to 0) For example: the set: ...
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A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
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### What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
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### Given $m^k\le n <m^{k+1}$ find $x$ and $y$ such that $x\cdot m^k+y=n$

Let $n,m,k\in\mathbb{N}$. Assume $m^k\le n <m^{k+1}$. Find $x,y\in\mathbb{N}$ such that (1) $x\cdot m^k+y=n$ (2) $0<x<m$ (3) $0\le y<m^k$ My question: does there exist a general ...
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### Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
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### Geometrical proof of the existence of square roots

This is quite an easy question, but it's been troubling me and I can't manage to work it out. I've been reading the book A Concise Introduction to Pure Mathematics (M. Liebeck), so I'll quote the ...
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### How to derive this formula about the bracket function?

Is there a direct way of proving that $$[nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
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### Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
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### How to calculate the number of lattice points in the interior and on the boundary of these figures with vertices as lattice points?

We define a point $(x,y)$ in the plane to be a lattice point if both $x$ and $y$ are integers. Now let $$S\colon= \{ (x,y) \ | \ 0 \leq x \leq m, \ 0 \leq y \leq \frac{nx}{m} \},$$ where $m$ and ...
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I have an ordered set with ten natural numbers in it such that when you sum up all ten numbers in the set, it is divisible by 7. Also, if you take the sum of the seven largest numbers in that set and ...
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### How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1}$ for $n \geq 2$, how to derive, without using induction, the inequality $$a_n < (\frac{1+\sqrt{5}}{2})^n$$ ...
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### Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
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### Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
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### Prove this simple arithmetic relation

Prove that if $$a \mid b$$ and $$a \mid c$$ then $$a \mid bx+cy$$ for any integers $x$ and $y$. Here's my proof: $$b = ak$$ $$c = am$$ $$bx+cy = akx+amy = a(kx+my)$$ Notice that $kx+my$ is an ...
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### 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?
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### The number $25!$ has exactly 7 trailing zeros, true or false?

I don't know how to determine it... any hints?
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### 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.
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### Determine an arithmetic relation

Let $f$ be an arithmetic function. Let $p$ be a prime number, $\chi(n)=\left(\frac{n}{p}\right)$ be a primitive Dirichlet character modulo p, where here $~\left(\frac{n}{p}\right)$ is the ...
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### Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$?

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$ for some $k\in \mathbb{Z}$ ? or we can prove that this never belongs to $\mathbb{Z}$ ?
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### Divisibility of the sum of a number and its 'mirror'

I came across the following puzzling problem in an elementary algebra textbook: Problem. Prove that the sum of a two-figure number and the number written with the same digits in the reverse order ...
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### proof by contradiction that if a and b are positive integars and $ab >100$ then at least one of the integars a and b is greater than 10 [closed]

does anyone know how to proof by contradiction that if $a$ and $b$ are positive integars and $ab >100$ then at least one of the integars $a$ and $b$ is greater than $10$
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### For any prime $p>3$ show that 3 divides $2p^2+1$

Does anyone know how to show this preferable without using modular For any prime $p>3$ show that 3 divides $2p^2+1$
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### Prove by induction that $a-b|a^n-b^n$ [duplicate]

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: ...
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### Number of digits of $2^{1000}$ [duplicate]

A friend asks me to find the number of digits of $2^{1000}$. I tried to look for a pattern by calculating the first powers of $2$ but I didn't find it. How should I proceed? Thanks.
### Proof $\displaystyle \text{lcm}(x,y)=\frac{|x\cdot y|}{\text{gcd}(x,y)}$ [duplicate]
Prove: $$\text{lcm}(x,y)=\frac{|x\cdot y|}{\text{gcd}(x,y)}$$ I used many ways to do it, all failed. One of them was to represent $|x\cdot y|$ as a sum of primes then $\text{gcd}(x,y)$ as a sum ...
### Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$.
Let's define $\mathbb{N}^*$ first: $\mathbb{N}^* = \mathbb{N} - \{0\}$ Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$. ...