# Tagged Questions

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### Find all the polynomials $p \in \mathbb R [X]$ such that $(x+1)p=(p')^2$

(Where $p'(x)$ is the derivative of $p(x)$) Research effort: what I thought is that given that $(x+1)|(p')^2$ then $(x+1)|(p')$ (I'd like to justify better this, but I don't know how) Then, ...
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### Let $a, b \in \Bbb Z$. Consider the following function: $f : ℤ \times ℤ \to ℤ$ such that for any$(x,y)\in ℤ \times ℤ, f(x,y) = ax + by.$

Let $a, b \in ℤ$. Consider the following function: $f : ℤ \times ℤ \to ℤ$ such that for any $(x,y)∈ ℤ \times ℤ, f(x,y) = ax + by.$ Fill in the blank in the following proposition with a simple ...
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### combinatorics and divisibilty

in how many ways we can form a $8$ digit numbers from $1,2,3,4,5$ with repetition allowed & divisible by $8$. MY APPROACH : to be divisible by 8 : last 3 digit of the no. must be divisible by 8 ...
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### Prove that $2^n3^{2n}-1$ is always divisible by 17

Prove that $2^n3^{2n} -1$ is always divisible by $17$. I am very new to proofs and i was considering using proof by induction but I am not sure how to. I know you have to start by verifying the ...
315 views

### if $X^2 +AX+B=0$ has a rational root, prove it is an integer. [duplicate]

If $x^2 + Ax +B =0$ has a rational root, prove it is an integer. I don't even know where to start on this problem. I've had it on my white board for about a week and keep looking at it, but can't see ...
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### Middle school number theory

Find at least three numbers that satisfy all three conditions: (1) there is a remainder of $1$ when the number is divided by $2$; (2) there is a remainder of $2$ when the number is divided by $3$; ...
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### Proving divisibility of numbers

Let us take a two digit number and add it to its reverse.We have to prove that it is divisible by 11. Same way,if we subtract the larger number from the other,it is divisible by 9.How can we explain ...
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### Find the greatest common divisor of the polynomials:

a) $X^m-1$ and $X^n-1$ $\in$ $Q[X]$ b) $X^m+a^m$ and $X^n+a^n$ $\in$ $Q[X]$ where $a$ $\in$ $Q$, $m,n$ $\in$ $N^*$ I will appreciate any explanations! THanks
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### Prove or disprove the following statements involving greatest common divisor

Help with prove or disproving either of these statements would be really appreciated, one or the other is fine, I just need a start or a solution to one and I'm sure I could probably figure the other ...
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### Proof involving division algorithm

I'm trying to prove the following. Let $m$ and $n$ be positive integers, $n>m$. Prove that if $n$ divided by $m$ leaves remainder $r$, then $2^n - 1$ divided by $2^m-1$ leaves remainder ...
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### How to prove that for all $m,n\in\mathbb N$, $\ 56786730 \mid mn(m^{60}-n^{60})$?

How to prove $\forall m,n\in\mathbb N$: $$56786730 \mid mn(m^{60}-n^{60})?$$ Thanks in advance.
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### Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$

Could you help me with the problem below? Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$. Thank you!
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### Divisibility problem: show $(x-z)\mid xy+zw \implies (x-z)\mid xw+yz$

I'm stuck at this homework problem can someone help me out? Much appreciated! $$(x-z)\mid xy+zw \implies (x-z)\mid xw+yz$$ Thanks again!
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### Problems with proof that $p|2^m-2^n$ if $p-1|m-n$

This was a homework assignment that I have already made unsuccesfully. However, no answers were given and I'm still curious. The question is as follows: "If $p$ is an odd prime number and $m > n$ ...
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### Is it allowed to divide an equation by an expression which can be equal to zero?

I need a help in such a problem and will greatly appreciate any suggestions. I was taught, division of an equation by an expression which can be equal to zero can lead to missing roots. But I thought ...
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### Finding the smallest positive integer $N$ such that there are $25$ integers $x$ with $2 \leq \frac{N}{x} \leq 5$

Find the smallest positive integer $N$ such that there are exactly $25$ integers $x$ satisfying $2 \leq \frac{N}{x} \leq 5$.
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### Solve for $px + q \equiv 0\pmod r$

How would I solve for the following in general: $(px + q)\equiv 0 \pmod r$ For example, $(2x + 1)=0 \pmod 7$ $x = 3, 10, 17, 24, \ldots$ $(9y + 5)= 0 \pmod 3$ $y$ has no ...
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### Prove $6 \nmid [\left( \sqrt[3]{28} - 3 \right)^{-n}]$

Prove that: $$6 \not\left|\ \left\lfloor\frac 1 {(\sqrt[3]{28} - 3)^{n}}\right\rfloor \ (n \in Z^+)\right.$$ ($\lfloor x\rfloor$ = largest integer not exceeding $x$) I am very bad as English and ...
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### Count the the number of elements in a set, exactly divisible by 2 out of 3 numbers

I need a hint to solve the following problem, in a way that a 10yr old child can understand. On a blackboard, all whole numbers from 1 to 2006 were written. John underlined all numbers divisible by ...
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### If an integer is divisible by 8 and 15, then the integer also must be divisible by which of the following?

I'm not going to list the choices here, mainly because I just want the general idea. If I generalize the question and was given $n$ different integers divide some integer $r$, how do I determine what ...
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### $S$ and $G$ are positive integers. Prove there exist integers $x$ and $y$ such that $x+y=S$ and $(x,y)=G$ if and only if $G\mid S$

So obviously because of the if and only if we must first prove that If there exist integers $x$ and $y$ such that $x+y=S$ and $(x,y)=G$ then $G\mid S$. And then if $G\mid S$, then there exist ...
### Suppose $p$, $q$ are distinct odd primes, $a\in\mathbb{Z}$, and $q|a^p-1$ but $q\nmid a-1$
From the assumptions above, I am trying to prove that $q=1+kp$ for some integer $k$ and that $k$ is even. My thoughts thus far: Since $a^p\equiv 1$ mod $q$, I know that by a corollary of Fermat's ...