# Tagged Questions

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### Prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$

Can we prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$ and if so, which formulas can be used while proving?
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### Show that $gcd(a,b) |d$ and hence $gcd(a, b) \leq d$, where $d$ is the smallest number of the form $ma+nb$

Show that if $d$ is the smallest element in the set $S = \{s \in \mathbb{N} | \exists m,n \in \mathbb{Z}, s = ma+ nb \}$ such that $d = ax + by$ then $\gcd(a,b) |d$ and hence $\gcd(a, b) \leq d$
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### What's the easiest way to factor $5^{10} - 1$?

What's the easiest way to factor $5^{10} - 1$? I believe $5 - 1$ is a factor based off the binomial theorem. From there I do not know. We are using congruence's in this class.
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### Define a recursive sequence for the following formula f(n) = n(n+1)

Define a recursive sequence for the following formula f(n) = n(n+1). Preferably one only defined by previous $a_n$ terms, i.e., no 'n' terms. If possible that is. So for example the following ...
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### Help with understanding Induction proofs

From my understanding, to prove induction problems, we must: Find a base case Assume n=k holds true Prove n=k+1 with the assumption However I am looking at the proof of a problem and they don't ...
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### Understanding a proof that $\gcd(a, b) = 1$ if $sa + tb = 21$ and $ua + vb = 10$

I am studying the solution to a problem: Suppose $a, b, s, t, u, v$ are integers such that $sa + tb = 21$ and $ua + vb = 10$. Show that $\gcd(a; b) = 1$. ...
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### how to prove if $m^{2}|n^{2}$ then m|n just hint please. [duplicate]

How to prove the following?: Let $m,n\in \mathbb{N}$ ; $\;m^{2}\mid n^{2}\Longrightarrow \;\;m\mid n$ Just a hint please. I tried two ways but did not work.
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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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### Prove that if $a$ is irrational then $\sqrt a$ is irrational

Just hints but solution thx. Any hints for me? I simply suppose that $a = \dfrac mn$ then $\sqrt a = \sqrt{\dfrac mn}$ But this does not make sense ..
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### $k$th difference of $1,2^k,3^k,…$

I read, in an exposition of Euler's proof of Fermat's theorem on sums of squares, that the $k$-th order finite forward difference of the function $f(x_i)=x_i^k$, relatively to the nodes $x_i=i$ where ...
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### decoding an encrypted text with modulo

A B C D E F G H I J K L M N O 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 P Q R S T U V W X Y Z Ä Ö Ü ß 16 17 18 19 20 21 22 23 24 25 26 27 28 29 00 A encryption method ...
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### Prove or disprove the following statement. $7 \ | \ (x^3 + x^2 + x + 2)$, where $x$ is an odd integer

We're learning about modulus and division (Discrete mathematics and proofs course). I'm not exactly sure how to tackle this sort of problem, is there some sort of property of cubic functions ...
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### showing that encryption method is bijective

A encryption method relates a letter Ω to letter $Δ\equiv aΩ + d$ $(mod 30)$ with $a, d\in {\Bbb N}$. Each letter relates to a number: A = 01, B = 02, C = 03 ... i. Show, that this encryption method ...
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### Greatest common divisor. Help [closed]

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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### How to find all the positive integral solutions of $5x+7y=100$?

How to find the number of all the positive integral solutions and the solutions itselves of $$5x+7y=100?$$ Please help me!!
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### Sum of floor of rational product

Given natural numbers $a,b,n$, where $a<b$, $n<b-1$, and $a$ and $b$ coprime, Find a closed form for the sum: ${\displaystyle \sum_{k=1}^n \left\lfloor k \frac{a}{b} \right\rfloor}$ We know ...
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### How many prime numbers are there in between $1000!+1$ and $1000!+1000$, inclusive?

I know $1000!$ is not a prime number as any number $1000$ or less is a divisor, but how would I know if $1000!+1$ is prime? Any hints? Also, use the above question to prove that you can find $n$ ...
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### $4011x+42053 \equiv 2x-782398 \pmod {10}$

$4011x+42053 \equiv 2x-782398 \pmod {10}$ $10|(4011x+42053-2x+782398) \space \rightarrow \space 10|(4009x + 824451)$ $\rightarrow\space 4009x\equiv -824451 \pmod {10}$ I am dubious about this next ...
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### Find the smallest positive $n$ such that $n! \equiv 0 \pmod {425}$

By exhaustion I found $n=17$, but in trying to solve this I can only see that: $$425\space |\space n!\space \longrightarrow\space 425k=n!$$ Any hints?
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### Prove $a,b \in \mathbb{Z}; p, q \in \mathbb{Z^+}$: If $a \equiv b \pmod{pq}\space \longrightarrow \space a \equiv b \pmod p$

$pq|a-b$ and $p|a-b$ $pql = a-b$ and $pk=a-b$ Let $k=ql$ then, $pql = pk=a-b$ First, is this correct? Second, if so, why can we let $k=ql$? How do we know $ql=k$?
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### Diophantine equation $11\cdot 2^y-3x^{10}=2014$

Ok so I have a trouble figure out here For the Diophantine equation $11\cdot 2^y-3x^{10}=2014$, either find all integer solutions, or show that there are no integer solutions.
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