# Tagged Questions

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### Show $4x^3 + y^3 = 792,864,313,578,917,724,246$ has no solution for $x, y \in \mathbb{Z}$.

I think it involves something about looking at the last digits of the number and/or modular arithmetic but I don't remember how to do this. Help?
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### Calculate the last digit of $3^{347}$

I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that 1st power last digit: 3 2nd power last digit: 9 3rd power last digit: ...
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### Find the remainder of $40^{314}$ divided by 91.

Here's what I have so far. $$x \equiv 40^{314} \mod{91}$$ $$\Rightarrow$$ $$x \equiv 40^{314} \mod{7}$$ $$x \equiv 40^{314} \mod{13}$$ Then by FLT, $$40^6 ≡ 1 \mod{7}$$ $$40^{12} ≡ 1 \mod{13}$$ ...
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### How to find $k$ in $6^{k} \equiv 2 \mod {13}$

Find for which $k$ is $6^{k} \equiv 2 \mod {13}$ I'm having trouble with these types of question in my cryptography class. This is part of Diffie–Hellman algorithm for calculating a shared key. ...
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### Divisibility question with 8th powers

so I was assigned a divisibility question for homework. Prove that $27195^8-10887^8+10152^8$ is divisible by $26460$. Am I supposed to use mods? I appreciate the help!
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### Show that if $N$ is an odd prime, then there are exactly $\frac{N + 1}{2}$ quadratic residues in the set $\{0, 1, …, N - 1\}$.

Fix a positive integer $N > 1$. We say that $a$ is a quadratic residue modulo $N$ if there exists $x$ such that $a \equiv x^{2} \pmod{N}$. Show that if $N$ is an odd prime, then there are ...
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### Show that there are exactly two values in $\{0, 1, …, N - 1\}$ satisfying $x^{2} \equiv a \pmod{N}$.

Fix a positive integer $N > 1$. We say that $a$ is a quadratic residue modulo $N$ if there exists $x$ such that $a \equiv x^{2} \pmod{N}$. Let $N$ be an odd prime and $a$ be a non-zero ...
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### Solve the following system of simultaneous congruences:

\begin{gather} 3x\equiv1 \pmod 7 \tag 1\\ 2x\equiv10 \pmod {16} \tag 2\\ 5x\equiv1 \pmod {18} \tag 3 \end{gather} Hi everyone, just a little bit stuck on this one. I think I am close, but I must be ...
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### How to Solve $x^a=b$ in $\mathbb{Z}/n$

I'm preparing for an exam by looking over old exams and have to exam questions of the same type but solved slightly differently. a) $x^{13} = 3 \bmod{47}$, given that $3^{37}=14$. (Answer: $x=32$) ...
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### Modular equation with very large powers

I studying for a discrete mathematics exam and am stuck on this question: Find the value of the unique integer $x$ satisfying $0 \le x < 17$ for which: $$4^{1024000000002} ≡ x \pmod{17}$$ I ...
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### Evaluate $\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2}$ where $\gcd(m,n)=1$

i have no clue on how to evaluate: $$\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2} \text{ where }D = \{ (m,n) \in (\mathbb{N}^*)^2 \mid \gcd(m,n) = 1\}$$ If someone is able to give me a ...
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### Math Parlor Trick

A magician asks a person in the audience to think of a number $\overline {abc}$. He then asks them to sum up $\overline{acb}, \overline{bac}, \overline{bca}, \overline{cab}, \overline{cba}$ and ...
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### Prove “casting out nines” of an integer is equivalent to that integer modulo 9

Let $s(x)$ be an abstraction for casting out nines of integer $x$. For all integers $x$, prove $s(x) \equiv x$ mod $9$ I'm not asking for an answer more of a way to attack this problem. Can't think ...
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### Prove that if $p_1,\dots,p_k$ are distinct odd primes then 1 has $2^k$ square roots $\mod m$ where $m$ is the product of the primes.

I think I am most of the way through this proof but I am stuck. Here was my approach: I looked at the square roots of $1$ mod $105$, and noticed that each one corresponded to one less than an integer ...
$4+2x≡7 \pmod 8$ Find all possible solutions and note any identities. Identify how you found the solutions. Explain what identities are.