Question

Find all solutions to $x^2 \bmod 15=1$ for $x \in\{1,2, \ldots, 14\}$.

    Find all solutions to $x^2 \bmod 15=1$ for $x \in\{1,2, \ldots, 14\}$.

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Applied Algebra: Codes, Ciphers and Discrete Algorithms

Darel W. Hardy, Fred… 2nd Edition

Chapter 7, Problem 5

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We need to find all integers \( x \) such that \( x^2 \equiv 1 \pmod{15} \) and \( x \) is in the set \(\{1, 2, \ldots, 14\}\). Show more…

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Find all solutions to $x^2 \bmod 15=1$ for $x \in\{1,2, \ldots, 14\}$.

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Key Concepts

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) is a tool that enables the decomposition of problems involving congruences with a composite modulus into several simpler problems with prime moduli. For cases like a modulus that is the product of distinct primes, one can solve the congruences modulo each prime factor and then combine the solutions to obtain the complete set of solutions in the original modular system.

Quadratic Residues

Quadratic residues are the elements of a modular arithmetic system that can be expressed as the square of some integer modulo n. Understanding which numbers are quadratic residues modulo a given number is central to solving quadratic congruences, as it informs whether a congruence has a solution and helps in determining the set of all solutions.

Modular Arithmetic

Modular arithmetic is the study of integers with respect to a fixed modulus, where numbers 'wrap around' after reaching a certain value. It is used to simplify computations by considering the remainders when integers are divided by the modulus, and is fundamental in many areas of mathematics, including number theory and cryptography.

Quadratic Congruences

A quadratic congruence is an equation where a quadratic expression, typically of the form x², is set congruent to a number modulo n. Solving such equations requires finding all integers x within a certain range that satisfy the given congruence. In many cases, this involves techniques like factoring or testing the possible residues.

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