Articles of modular arithmetic

How to find $\ker T$ from matrix representation over polynomial finite field?

Let $\mathbb F$ be a finite field of 5 elements (essentially $\mathbb Z_5$). $T:\mathbb F_3[x] \to \mathbb F_3[x]$ is a linear map defined by the representation matrix: $$ [T]_B=\begin{pmatrix} 1&2&3\\ 1&0&4\\ 0&1&2 \end{pmatrix} $$ with the basis $B=(1, 1+x,1+x+x^2)$. Find $\text{ker}T$. $\mathbb F_3[x]$ is a space of polynomials of form $ax^2+bx+c$ over $\mathbb F$. We […]

modulus calculations & order of operations

This is a 2 part question. part 1 (negative mod calculations): As part of a larger equation, I have come to a stage where I need to calculate -17 mod 11. By doing it manually I got -6 as the result. (-17 – ((-17 / 11) * 11)) But by checking an online mod calculator, […]

How to find modulus square root?

Given integers $m$, $c$ and $n$. Find $m$ such that $m^2 \equiv c \ (\mod n) $ I used Tonelli-Shanks algorithm to caculate the square root, but in my case $n$ is not a prime number, $n = p^2,\ p$ is a prime number. I read this page. It is said that: In this article […]

Show x mod p is a cubic residue iff $x^{(p-1)/3} \equiv 1$ mod p.

$x$ is a cubic residue mod p if it is of the form $a^3$ mod p for some residue $a$. Show if $p\equiv 1$ mod 3, then x mod p is a cubic residue iff $x^{(p-1)/3} \equiv 1$ mod p. Also, show if $p\equiv 2$ mod 3, then all x mod p are cubic residues. […]

Reverse of Chinese Remainder Theorem

For the following: $(102n-51) \not\equiv 2 \pmod {2,3,5,7,11,13,…,\sqrt{102n-51}}$ (That’s probably completely incorrect use of symbols, but I mean not equivalent to 2 mod any prime less than $\sqrt{102n-51}$) here are my questions: What is the first (smallest) $n$ solution? Are there infinitely many $n$ solutions? (Most importantly) Is there a way we’d know (be able […]

Modulo with negative numbers

When I input this in wolfram I get false -347 mod 6 = 5 When I input this I get true -347 mod 6 = 1 And yet I know $-5 \equiv 1$ And additionally $-6*57 – 5 = -347$ but $-6*57 – 1 \neq -347$ So it’s strange that Wolfram’s answer is true for […]

Negative modulus

In the programming world, modulo operations involving negative numbers give different results in different programming languages and this seems to be the only thing that Wikipedia mentions in any of its articles relating to negative numbers and modular arithmetic. It is fairly clear that from a number theory perspective $-13 \equiv 2 \mod 5$. This […]

Find last two digits using modular arithmetic

This question already has an answer here: Find the last $2$ digits of $7^{7^{7^{10217}}}$ [duplicate] 1 answer

Prove that :- $A_n = 2903^n – 803^n – 464^n + 261^n$ is divisible by $1897$ for $n \in \mathbb{N}$

Prove that :- $A_n = 2903^n – 803^n – 464^n + 261^n$ is divisible by $1897$ for $n \in \mathbb{N}$ I tried induction but got no where :(. $p(n)$ : $A_n = 2903^n – 803^n – 464^n + 261^n \mod 1897 = 0 $ for $p(1):$ the statement is true as $A_1 = 1897 \mod […]

How do I find the period of repetition of this modular equation?

EDITED QUESTION: I have this equation $$\left(x+\frac{2}{3}\right)\,mod\, \frac{13}{5}$$ and I want to find period after which its value repeats. It can be written as $$\left(\frac{3x+2}{3}\right)\,-\, \frac{13}{5}q = 0$$ I am equating it to some number (here I am equating it to zero to make the calculations simple), in order to find its period. And $q$ […]