# Wolfram alpha congruence modulo

Here is the "proof" of my counterexample, which seems to be too large to compute directly (crashed the Sage program and Wolfram didn't understand it directly, so more work was needed).

Knowledge-based, broadly deployed natural language. Wolfram Science. Technology-enabling science of the computational universe. Wolfram Natural Language Understanding System.

Fast Modular Exponentiation. Modular inverses. … 05/03/2021 10/03/2021 Mod[m, n] gives the remainder on division of m by n. Mod[m, n, d] uses an offset d. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

## Added May 29, 2011 by NegativeB+or-in Mathematics. This widget will solve linear congruences for you. The equation 3x==75 mod 100 (== means congruence), input 3x into Variable and Coeffecient, input 100 into modulus, and input 75 into the last box.

Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org » Join the initiative for modernizing math education.

### If I need to find the solutions of: x^2 ≡ 8 (mod 23) and I use the above formulas, I get the solution: x = +- 13, but Wolfram Alpha says that +-10 also works (and it does). So why is that? Why do these formulas only give you 1/2 of the answer and why can't I find a way to find the whole answer (excluding the way where you draw the table with x^2 mod p for x from 0 to p and)? I realize that I'm probably doing …

The code I have is Table[If[{ModularInverse[i, 120 ]} = {}, 120, ModularInverse[i, 120]], {i, 0, 119}] If the modular inverse does not exist, it should return $120$. If it does exist, it should return the integer … Here is the "proof" of my counterexample, which seems to be too large to compute directly (crashed the Sage program and Wolfram didn't understand it directly, so more work was needed). This package implements the Gauss-Lagrange algorithm to find the canonical form under congruence of a symmetric matrix associated with a real quadratic form. This allows one to classify all real quadratic forms, and in particular to determine whether a given quadratic form is positive definite or not. The package also implements elementary row and column operations on any matrix.

Wolfram|Alpha can recognize millions of Congruence. If two numbers and have the property that their difference is integrally divisible by a number (i.e., is an integer), then and are said to be "congruent modulo ." The number is called the modulus, and the statement "is congruent to (modulo )" is written mathematically as Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Added May 29, 2011 by NegativeB+or-in Mathematics. This widget will solve linear congruences for you. The equation 3x==75 mod 100 (== means congruence), input 3x into Variable and Coeffecient, input 100 into modulus, and input 75 into the last box. To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: To add a widget to a MediaWiki site, the wiki must have the Widgets Extension installed, as well as the code for the Wolfram|Alpha widget . Wolfram|Alpha » Explore anything with the first computational knowledge engine.

Solve systems of congruences: Solve a congruence involving variables in the modulus: Solve systems with each equation under a different modulus:. Mod is also known as modulo operation. · Mathematical function, suitable for both symbolic and numerical manipulation. · Typically used in modular arithmetic,  are said to be "congruent modulo m Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic Solve integrals with Wolfram|Alpha .

The solution of a linear congruence can be found in the Wolfram Language using Reduce[a*x == b, x, Modulus -> m]. Solution to a linear congruence equation is equivalent to finding the value of a fractional congruence, for which a greedy-type algorithm exists. In particular, (1) can be rewritten as Mar 10, 2021 · Modular arithmetic is the arithmetic of congruences, sometimes known informally as "clock arithmetic." In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity, which is known as the modulus (which would be 12 in the case of hours on a clock, or 60 in the case of minutes or seconds on a clock). Wolfram Science. Technology-enabling science of the computational universe. Wolfram Natural Language Understanding System.

I notice that 777=259(3) but I don't think I'm allowed to just multiply the modulus by 3 to make it work. Any ideas? Since I'm not going to be tested on this, I can just ask Wolfram Alpha: ChineseRemainder[{2, 5, 1}, {3, 7, 8}]. Besides, the steps to solving simultaneous congruences are covered in other Math.SE questions and answers. Wolfram Alpha tells me the answer is 89. It checks out: $$89 = 3 \times 29 + 2$$ $$89 = 7 \times 12 + 5$$ 89 = 8 \times 11 Goldbach's conjecture [], formulated in 1742 and still unproven, asserts that system is equivalent to True.This suggests that the Wolfram Language may not be able to solve arbitrary Diophantine polynomial systems.

$\begingroup$ $11$ is a prime, so integers modulo $11$ form a field. This means that you can use the same techniques as is taught over the reals in Linear Algebra. Form the matrix, and use elementary row operations. See my old answer for a walk-thru example of inverting a matrix modulo $29$. The multiplicative inverse of a modulo m exists if and only if a and m are coprime (i.e., if gcd(a, m) = 1). To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. The Wolfram Language uses a variant of the modular sieve method (see e.g.