# Tagged Questions

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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### Prove that $2^n$ does not divide $n!$

I want to prove that $2^n$ does not divide $n!$. I was trying by induction and I'm confused about if what I'm doing is right. First I test it with $n=1$. In fact: $$2^1 \nmid 1!$$ So if i take the ...
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### If $a \in A$ and $b \in B$ then $2a \in B$ and $2b \in A$ and $(a+b)^{2014}\in C$ [closed]

Below are questions that it think I know how to do but im not $100\%$ sure. $(i)$ asks if $a$ is odd so $a=k+1$, then prove $2a$ is even so $2a = 2k+2.$ The second and third differ a little am I ...
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### Divisibility of a summation by $p^2$

I try to use the hint of this problem but I could not. Any detailed answer will be appreciated! Let $p$ be a prime number which $p>3$, and $$a/b:=1+1/2+1/3+\cdots +1/(p-1).$$ How could we show ...
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### Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$

I'm stuck with this exercise: Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$ It's from my algebra class, we are looking into diophantic and congruence equations. ...
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### Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $m$ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart ...
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### Proof for any natural n that: $8|5^n+2*3^{n-1}+1$

I used this method for proving this statement but I came up with a problem. $5^n+2*3^{n-1}+1 \equiv 1 + 25^{n/2} + 2 * 81^{(n-1)/4} \equiv 4 \pmod{8}$ What is the problem with my solution?
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### Rayleigh quotient $Q=(\frac{||\triangledown w||}{||w||})^2$ in using the eigenfunction $\sin(x)$ on the segment $(0,\pi)$

I would like to well understanding the Rayleigh quotient $Q=(\frac{\|\nabla w\|}{\|w\|})^2$. Does anyone could explain to me why we divide the norm of the gradient $\| \nabla w \|$ by $\| w \|$, and ...
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### Find all odd $n \in \mathbb{Z}^+$ such that $n\mid 3^n+1$.

Find all odd $n \in \mathbb{Z}^+$ such that $n\mid 3^n+1$. I believe that there doesn't exist any such $n$ except $1$. It is clear that $n$ can't be a multiple of $3$. Also, $3^n \equiv -1 \pmod n$. ...
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### Proving that an equation doesn't have integer solutions

I need to prove that there are no integer solutions for a bunch of equations like the following: $$15x^2 - 7y^2 = 9$$ I was able to solve some simpler ones by picking a dividend and looking into it's ...
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### Show there are only a finite number of integers with $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)}$ an integer

Show, for each $n$, there are only a finite number of integral $(a_i)_{i=1}^n$ such that $2\le a_i \le a_{i+1}$ and $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)}$ is an integer. My question is ...
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### Divisibility - what is A+B?

Is there an easy to solve this problem? I can find the answer by using a complicated rule that I don't understand. Even if I try to remember this rule, I probably will forget about it a year later. ...
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### Find the remainder of $\sum_{i=0}^{99} 2^{i^2}$ when dividing by 7 and determine if the quotient is even or odd

I've recently had this problem in an exam and couldn't solve it. Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd: $$\sum_{i=0}^{99} 2^{i^2}$$ ...
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### Problem to find all $n$ in following situation [closed]

Find all $n>1$ such that $1^{n} + 2^{n} + 3^{n} +\cdots + (n-1)^{n}$ divisible by $n$. I'm not good at Number Theory so , give elementary answer.
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### Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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### Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By brute-...
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### Show that among any consecutive $16$ natural numbers one is coprime to all others

Show that among any consecutive $16$ natural numbers one is coprime to all others. Is it useful to use the division algorithm on $16$? $16k,16k+1,16k+2,...16k+15$
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### How Euclidian Algorithm for division works with algebric expressions?

I am attending an introductory Number Theory class for Computer Science focused on cryptography. I have done some exercises with integers number but I have two exercises in which appears algebric ...
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### Finding quotient and remainder for a division

We are starting with division and congruence in my algebra course... this is one of the first exercises for the division algorithm. I've done the first that were given with fixed values but now I have ...