Tagged Questions

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An easy question with integer numbers

I have an easy question of arithmetic. Let $a, b, N$ be integer numbers such that $\mathrm{gcd}(a,b,N) = 1$. Is it true that there exists an integer number $x \in \mathbb{Z}$ such that ...
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99 Ninja missions and a single lantern

From http://blog.liveramp.com/2014/01/30/the-case-of-ninety-nine-ninja/: A team of 99 ninja are each sent out on individual missions by their master. Their master tells the ninja that one of them ...
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Greatest common divisor. Help

Let $n \in \mathbb{N}$. Prove that $$\gcd(2^n+7^n;2^n-7^n)=1$$ $$\gcd(2^n+5^{n+1};2^{n+1}+5^n)=3\text{ or }9$$
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Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
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Solving $[x]+[x]=[2x]$

Solving the equation $[x]+[x]=[2x]$ Since $[x]$ is the greatest integer function. I tried, $\forall x\in\mathbb{N}$, we have $[x]=x$ and $[2x]=2x$ this implies that $[x]+[x]=[2x]$, but if ...
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Sum of the reciprocals of divisors of a perfect number is $2$?

How do I show that the sum of the reciprocals of divisors of a perfect number is $2$? I tried $d_i\mid n$ with $i\in\mathbb{N},\;d_i\leq n$ then ...
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Find $a,b \in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$

$1$. Find $a;b\in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$ $2$. Find $m;n>1$ such that : $2^m+3^n=k^2$ $(k\in \mathbb{Z})$ Problem 1. I thought : ...
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A problem in fractions from a very old arithmetic textbook

Similar in vein to a problem I posted before here, I would be interested if anyone can give me any pointers as to how one might solve this question from the same arithmetic textbook: "Simplify ...
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How to find $p$ and $q$ if we have $\operatorname{lcm}(p,q)=b$ and $p+q=a$ where ($a,b \in \mathbb{N}$) and $p>q$.

What is the general formula to find $p$ and $q$ if we have $\def\lcm{\operatorname{lcm}}\lcm (p,q)=b$ or $\gcd(p,q)$ and $p+q=a$ where ($a,b \in \mathbb N$) and $p>q$? Example: $\lcm(p,q)=84$ and ...
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How to explain the commutativity of multiplication to middle school students?

It's easy for natural numbers: $3\times 5=5\times 3$ ***** ***** ***** but how do you explain that $x.y=y.x$ for any real numbers $x$ and $y$. Moreover, in $\Bbb{N}$, do you prefer to define ...
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Criterion for sum/difference of unit fractions to be in lowest terms

Pick two nonzero integers $a$ and $b$, so $(a,b)\in (\mathbb{Z}\setminus\{0\})\times(\mathbb{Z}\setminus\{0\})$. We want to add the fractions $1/a$ and $1/b$ and use the standard algorithm. First ...
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Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
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Correlation between multiplied numbers?

I do not have a strong math background, but I'm curious as to what this pattern is from a mathematical standpoint. I was curious how many minutes there were in a day, so I said "24*6=144, add a 0, ...
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A proof of $n*0=0$?

The only proof I've seen for this assumes that $0$ follows all the rules of arithmetic. How can we make that assumption when dividing by $0$ is a problem? I know that some people don't agree that all ...
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How many solutions are there to $abc+def=ghi$, where $a,b,\ldots, h,i$ are distinct non-zero digits?

I saw this problem posted by Google. Those posting in the comments found solutions using computer programming. I would like to know if there is an easier solution than trying every single combination. ...
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Are the propreties of arithmetic unproven?

For example, the property which says that $$a(b+c)=ab+ac$$ This is very clear for integers, but is it actually provable for all real numbers (and complex maybe). Or the commutative property which says ...
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Looking for name of theorem: “rational $\Leftrightarrow$ fractional part terminates or repeats”

I am looking for the name of the theorem that says that a number $x$ is rational if and only if its fractional part terminates or repeats (where "fractional part" refers to the representation of $x$ ...
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How to factor 5671?

The other day I wanted to factor 5671 in my head. (It turns out to be $53\cdot107$, but I did not know this at the time.) I quickly ruled out the easy divisors, 2, 3, 5, 7, 11, and 13. At this point ...
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$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff}\gcd(a,b)=1$

$$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff} \gcd(a,b)=1$$ This seems almost painfully obvious because it is just $ar+bs=1$ in another form. This second form is the definition of ...
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how to show that a function$f$ is contained in all natural numbers?

Let $f:\mathbb{N}\times\mathbb{N}\rightarrow \mathbb{R}$ be defined by $f(a, b) = \frac{(a+1)(a+2b)}{2}$. Carefully show that the image of $f$ is contained in $\mathbb{N}$.
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How do I append an integer to the left of another integer?

For example: . is my append operator f(x,y) = |x| . |y| f(1,45) = 145 f(233,10) = 23310 f(8,2) = 82 f(0,1) = 1 This is a trivially easy problem to ...
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Prove that $2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$

I am utterly new to modular arithmetic and I am having trouble with this proof. $$2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$$ It's because $2+5=3+4=7$, but it's not so clear for me ...
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A simple dividation problem

I can solve the below question by putting the each possibility into the questions conditions but I want to find out the value without putting the possibility . The number m yields a remainder p when ...
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Constrictions on A.P with factorials.

There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression. Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
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Don't understand casting out nines

Let n be a positive integer. If the sum of the digits of n is divisible by 9, then n is divisible by 9. I got upto here, ...
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Number base conversion

How can I convert a number from one base, $b_1 \neq 10$ to another base $b_2 \neq 10$ without going through base $10$ i.e. $b_1\rightarrow 10 \rightarrow b_2$?
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$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
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selecting an arbitary digit from an integer

Let us say I have an integer of an arbitrary length such as: $209484250490600018105614048117055336$ Is there an elegant function which allows me to select the $n$-th digit such that: $f(1) = 6$ ...
159 views

Is the length of a segment between 0 and 1 exactly 1?

What is the length of the line segment between points A and B on a number line, where A = 0 and B = 1? Is it exactly 1? Perhaps I am thinking about it in an incorrect manner, but it seems to me that ...
377 views

How many powers of 2 are easy to double? [duplicate]

Possible Duplicate: Is 2048 the highest power of 2 with all even digits (base ten)? Numbers written in base $10$ are easiest to double when their digits lie in the range $0, \ldots, 4$, so ...
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Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz

The question is motivated by a question on a can of number noodles. Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
Prove that given any rational number there exists another greater than or equal to it that differs by less than $\frac 1n$
Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...