- In Exercises 71 to 82 , find all whole number solutions of the congruence equation. (2 x+2) ≡6

Step 1: Start with the given congruence equation: u000Au005C[u000A(2x + 2) u005Cequiv 6 u005Cmod 4u000Au005C]u000Au000AStep 2: Simp

- In Exercises 71 to 82 , find all whole number solutions of...

Key Concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value called the modulus. It involves operations like addition, subtraction, and multiplication where results are taken modulo a fixed number, which organizes integers into equivalence classes.

Congruence Relations

A congruence relation states that two integers are equivalent with respect to a modulus if their difference is divisible by that modulus. This is denoted as a ? b (mod m), meaning that m divides (a - b), and it provides a foundational tool for working in number theory and modular arithmetic.

Linear Congruence Equations

Linear congruence equations are equations in the form ax ? b (mod m) where the goal is to find all integer solutions for x. Solving these equations typically involves simplifying the congruence, checking for the existence of solutions using divisibility conditions (such as whether the greatest common divisor of a and m divides b), and then expressing the general solution as a set of integers that satisfy the congruence.

Solution Sets Representation

Once a linear congruence has been solved, its solution set is usually expressed in terms of a residue class. This means that all solutions can be described by a single particular solution plus an integer multiple of a certain number, often derived from the modulus or its factors, thereby capturing all whole number solutions systematically.

- In Exercises 71 to 82 , find all whole number solutions of the congruence equation. $(2 x+2) \equiv 6 \bmod 4$

Instant Answer

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