# Tagged Questions

For questions about congruences in modular arithmetic, concerning for example the chinese remainder theorem, Fermat's little theorem or Euler's totient theorem.

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### For all $x$ , $x^2 \equiv 0$ or $1$ or $4 \mod 7$

My textbook makes the following claim For any $x$ , $x^2 \equiv 0$ or $1$ or $4 \mod 7$ I can't see how this true though. $3^2 \equiv 4^2 \equiv 2 \mod 7$ so this obviously doesn't fall into ...
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### $i \equiv k \mod p \implies i = k$ if $p$ is prime?

In a particular proof of Fermat's Little Theorem $\big(a^{p} \equiv a \mod p \big)$ in Engel, the following fact is used $i \equiv k \mod p \implies i = k \:$ where $p$ is a prime. I'm not really sure ...
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### How many solutions of $\mod 63$ : $x^2=1 \pmod7$ and $x^3=1\pmod 9$ [on hold]

How many solutions $\mod 63$ , we have for: $$x^2=1 \pmod 7$$ and $$x^3=1 \pmod 9$$ Need to find them also.
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### Why does an even $x$ imply $y^2=-2 \pmod 8$

I am very new to modular arithmetic, and I encountered the following statement on page 7 of this paper: If $x$ is even then $y^2 \equiv-2\pmod{8}$ The equation in question is $y^2=x^3-2$ I do not ...
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### Prove a linear congruence equation [closed]

Let $p$ be a prime number. Prove that $2(p-3)! ≡ -1\text{ (mod } p)$.
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### For which $0\leq a<p^2$, where $p$ is an odd prime, we have that $(2p-1)!\equiv a\mod{p^2}$

Let $p$ be an odd prime. I need to find for which $0\leq a < p^2$, $(2p-1)!\equiv a\mod{p^2}$. If $a\equiv (2p-1)!\mod{p^2}$, then we have that $a = kp^2 + (2p-1)!$, and therefore $p\mid a$, ...
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### Solving linear congruences with unknown modulus

I need to resolve the following system of linear congruences: $9 = 3a+c \pmod m$ $11 = 9a+c \pmod m$ $1 = 11a+c \pmod m$ How can I proceed? Is there anyway to input these equations in Wolfram ...
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### Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$.

Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$. I'm having a difficult time proving this problem. I was able to verify that it works for prime $n$ up to ...
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### Construct a function pertaining to the OEIS sequence A131229 (Numbers congruent to {1,7} mod 10)

OEIS sequence A131229 ("Numbers congruent to {1,7} mod 10") begins $\{1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51,...\}$. I want a function $f(x)$, specifically such that $f(\frac{1}{2}) =\frac{7}{2}$, ...
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### The difference between congruence and equivalence class?

I've got an excercise solved by my teacher, it says I've got to prove a relation $R$ of elements in $\mathbb{R}^2$ is a congruence. In the solved exercise he just proved Reflexivity, transitivity and ...
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### Help with finding the remainder of $2^{2^n}$ when divided by 13

I have this problem from an algebra course: Find the remainder of $2^{2^n}$ when divided by 13, $\forall n \in \Bbb N$ It's in a section of Fermat's little theorem and Chinese Remainder Theorem ...
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### Let $a \in \Bbb Z$ such that $gcd(9a^{25}+10:280)=35$. Find the remainder of $a$ when divided by 70.

I'm stuck with this problem from my algebra class. We've recently been introduced to Fermat's little theorem and the Chinese Remainder Theorem. Let $a \in \Bbb Z$ such that $gcd(9a^{25}+10:280)=35$...
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### How do I solve this congruence?

I have some difficulties solving the following congruential equation. $3n^2 + 2 ≡ 0\pmod 5,\ \forall\ n \in Z$ If I subtract both members by $-2$, I end up getting $3n^2 = -2\pmod 5$ and I can't ...
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### Quadratic residue $p \equiv 1 \pmod 4$

Suppose $p$ is a prime congruent to $3$ modulo 4. Additionally, suppose $a$ is a quadratic residue modulo $p$. Prove that $x=a^{\frac{p+1}4}$ is a solution to the congruence $x^2\equiv a \pmod p$ ...
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### Prove the congruence $\sum_{r=1}^{p-1}{(r|p) * r } \equiv 0 \pmod p.$

Prove that if $p$ is prime and $p\equiv 1 \pmod4$, then $$\sum_{r=1}^{p-1}{(r|p) * r } \equiv 0 \pmod p.$$ ( $(r|p)$ is a Legendre Symbol ) I know that $\sum_{1 \le r \le p}{(\frac{r}{p})} = 0$, but ...
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### -3 is quadratic residue if and only if $p \equiv 1,7 \pmod {12}$ [closed]

I have to prove that -3 is quadratic residue if and only if $$p \equiv 1,7 \pmod {12}$$ I know one method (with symbol Legendre'a) but I don't get. If someone can explain me I will be happy or give ...
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### Prove: If $p|2^q -1$, p,q primes numbers then p is congruent to 1 modulo q

p|2^q -1, so 2^q - 1 = pk and 2^q = pk - 1 from Fermat we've got 2^q is congruent to 2 (mod q) pk-1 is congruent to 2 (mod q) pk is congruent to 1 (mod q), then k must be 1 Is this evidence ...
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### Computing difference in modular arithmetic. [closed]

Is there a meaningful kind of difference "$|a-b|$" in modular arithmetic? For example, in mod $12$, we would like to have $|0-11|= 1$ and $|0-1| = 1$.
### What is the Least Prime Factor of $3^{3241} + 8^{2433}$
I'm not sure how to do this question Attempt $$3^{3241} + 8^{2433}$$ I start by taking this number mod 3 $$3^{3241} + 8^{2433} \equiv 8^{2433} \mod 3$$ No we can see that $8^2 \equiv 1 \mod 3$. So ...