# Tagged Questions

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $b-a$.

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### Application of Fermat's Little Theorem/Fermat Euler Theorem

Find an integer $x$ with $0\leq x \leq73$ such that $$2^{75}\equiv x \pmod{74}$$ I think I'm supposed to be using either Fermat's Little Theorem or the Fermat-Euler theorem here but I don't think I ...
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### On exponent mod $2p$.

Assume $p$ is a prime. Assume $g$ is primitive root for both $\Bbb Z_p$ and $\Bbb Z_{2p}$. We know in discrete logarithm problem $z$ is unique $\bmod(p-1)$ in $g^z=h\bmod p$. Is it true that $z'$ ...
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### Equality simplification — have I done it correctly?

I previously posted this equality and got some nice feedback. This is my final equality to prove Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures. The first equality was to determine if ...
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### Prove: $\bar{a}^2 = \bar{0}$ in $\mathbb{Z}_{pq} \rightarrow \bar{a}=0$ where $p\neq q$ are primes

For this summer, I am teaching myself abstract algebra and I've been working on a proof for the following statement. I just need someone to confirm whether it is sound. (Note: Here, $\bar{a}$ denotes ...
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### Calculating modulus by coprimes

I need to calculate $x$, which is defined as the unique integer in $\{0,1,...,(pq - 1)\}$ such that $x \equiv n\mod{pq}$, where $n > 0$, $p$ and $q$ are relatively primes ($n, p, q$ are known ...
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### How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
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### How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
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### Proving the element of a symmetry group $\sigma^i \in S_n$ is of order $n$ and length $n$ only when $(n,i) = 1$

Start with element of $S_n$ as $\sigma^i$ which permutes an element of the set $\{1,2,3,...,n\}$, call it, $a_k \to a_{k+i}$ So $({\sigma^i})^2$ would permute $a_k \to a_{k+2i}$ If $k+i > n$, the ...
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### Systems of linear equations in the same modulus

I am working with a system of linear equations all taken with the same modulus, $n$, there is no assumption on $n$ other then it is at least 3 (really don't want to assume it is prime) I don't have ...
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### Find the remainder when a large number is divided by 35.

I don't know why I am wrong with this problem. Here is what I did: The last two digit of $6^{2006}$ is 36. So the answer should be 1. Find the remainder when $6^{2006}$ is divided by 35.
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### Number of roots of quadratic polynomial in $Z/(pq Z)$

I want to prove that quadratic polynomials in $Z/(pq Z)$ have at most 4 roots, when $p, q$ are prime. I currently do this by factoring the polynomial, $(x-a)(x-b)$ and then showing that either x ...