# Tagged Questions

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### A question of divisibility.

Let $p$ and $q$ are relatively prime integers. Consider $S = \{\frac{p}{q}\} + \{\frac{2p}{q}\} + \{\frac{(q-1)p}{q}\}$, where $\{x\}$ is the fractional part of the real number $x$. Prove that $2S$ ...
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### $12345679$ and friends

We can see that in the decimal system each of $12345679\times k$ $(k\in\mathbb N, k\lt 81, k\ \text{is coprime to$9$})$ (note! not $123456789$) has every number from $0$ to $9$ except one number as ...
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### Prove that any two numbers of the form $2^{2^n}+1$ are coprime to one another.

Full problem statement: Prove that any two numbers of the follwing sequence are relatively prime: $2 + 1, 2^2+1, 2^4 + 1, 2^8+1, ... 2^{2^n} + 1$ So far I have tried to use Euclid's algorithm with ...
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### Any other prime numbers that satisfy this condition?

If $a=2$ and $b=3$, then $a^2-1$ is an integer multiple of $b$. Is there any other pair of primes $a$ and $b$ that satisfy this relationship? I don't think so, but can't figure out why not.
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### Is there an algebraic method to concat two numbers?

I'm searching an algebraic way to concat numbers in base $10$. Concatening two numbers is to put side by side their notations. Let $c$ a concatenating function. $c(2,2) = 22$ $c(8,9) = 89$ ...
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### Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
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### Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
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### How many zeros are there in $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$?

How many zeros are there at the end of $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$ ? I tried it by grouping all the $2$'s and $5$'s and $5$'s and $6$'s but cant get my answer...
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### gcd finding method

An integer $d$ is a $\gcd$ of two non-zero integers $a$ and $b$, if $d$ divides $a$ & $d$ divides $b$ '$c$ divides $a$ & $c$ divides $b$' implies '$c$ divides $d$' for any integer $c$. If ...
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### Can $262537412640768743.99999999999925$ be beaten with simple expressions? [closed]

We know: \begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align} where $x$ is the real solution ...
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### Minimum number of coconuts

Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the ...
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### Distribution of palindromic numbers

We all know what a palindromic number is, it is a number which is the same, independent from which side we read it, for example 101, 202, 33733,.... It is also clear that there are infinity many ...
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### Recreational number theory problem

Suppose we have a positive integer $n$ that has exactly three distinct prime factors, say $p,q, r$. How can we find a formula for the number of positive integers $\leq n$ that are divisible by none of ...
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### Number theory Exercise

for positive integer $n$, how can we show $$\sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)}$$ where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)}$ if $n$ is square ...
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### Do all natural numbers have a nonzero multiple that is a palindrome in base 10?

Some natural numbers have a nonzero multiple that is a palindrome in base 10. For example, $106 \times 2 = 212$, which is a palindrome, and $29 \times 8 = 232$, which is also a palindrome. Aside ...
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### How to calculate $1^k+2^k+3^k+\cdots+N^k$ with given values of $N$ and $k$? [duplicate]

Here $1<N<10^9$ and $0<k<50$ So we have to calculate it in order of $O(\log N)$.
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### Primes in a certain arithmetic progression

Primes $=_m 1$. For any positive integer m, prove that arithmetic progression $$1 +m, 1 + 2m, 1 + 3m, ...$$ contains infinitely many primes. How can we prove that it suffices to show that for ...
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### Triangle from a given rectangle

We are given a set of (marked) points in a 2D coordinate system and function $f(x,y)$ which counts number of points marked in the rectangle $(0 , 0), (x , y)$ - where $(0 , 0)$ if down-left corner, ...
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### Does $p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n)$ imply $f(m+n)=f(m)+f(n)$?

Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that: $$(\forall p: \mathrm{~prime~})(\forall m,n\in\mathbb{N})(p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n))$$ is $f$ linear? by linear I mean: ...
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### Are numbers with repeating patterns in their decimal expansion (e.g. $0.123123123\ldots$) rational?

There's a question that I've been thinking about for quite some time now. We all know that numbers with infinite decimal expansion such as $0.\overline{3}$ or $0.\overline{1}$ are not necessarily ...
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### Are $4ab\pm 1$ and $(4a^2\pm 1)^2$ coprime?

Let $a\ne b$ be two positive integers. Are $4ab+1$ and $(4a^2+1)^2$ coprime always? Can you find $a$ and $b$ such that they are not coprime? Edit: It has been proved that $4ab-1$ is not a divisor ...
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### Number theory fun problem

Say $a,b > 2$ are integers. Then we have that $2^a + 1$ is not divisible by $2^b - 1$. Any thoughts on how to tackle this problem???
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### All positive integers satisfying $n2^n=5^m+7$

How can one solve each of the equations below in positive integers? $$2^n=5mn+7$$ $$mn2^n=5^m+7$$ $$n2^n=5^m+7$$
### Constructing Magic Squares over $\mathbb{Z}$ from Magic Squares over $\mathbb{Z}_m$
A magic square over $\mathbb{Z}$ is an n x n matrix whose entries are $\{1, \ldots, n^2\}$, with the sum of every row and column identical (in particular, my magic squares are all normal, but the sum ...