Math Problem Statement

Fin

Find the whole number solutions of the congruence (2x + 3) ≡ 3 mod 4.

Solution

The image shows a problem asking to "Find the whole number solutions of the congruence 2x+33mod42x + 3 \equiv 3 \mod 4." The options listed are:

  • A) {0,3,7,11,15,}\{0, 3, 7, 11, 15, \dots \}
  • B) {1,5,9,13,17,}\{1, 5, 9, 13, 17, \dots \}
  • C) {2,6,10,14,18,}\{2, 6, 10, 14, 18, \dots \}
  • D) {3,8,12,16,20,}\{3, 8, 12, 16, 20, \dots \}

Let's solve this congruence step-by-step:

Step 1: Solve the congruence

We are given the equation: 2x+33mod42x + 3 \equiv 3 \mod 4 Subtracting 3 from both sides: 2x0mod42x \equiv 0 \mod 4 This simplifies to: x0mod2x \equiv 0 \mod 2 which means that xx must be an even number.

Step 2: Check the solutions

xx can take values 0,2,4,6,0, 2, 4, 6, \dots, so we need to check which of the given options contains these numbers.

Looking at the options:

  • A) {0,3,7,11,15,}\{0, 3, 7, 11, 15, \dots \} includes 0 but also odd numbers.
  • B) {1,5,9,13,17,}\{1, 5, 9, 13, 17, \dots \} contains odd numbers only.
  • C) {2,6,10,14,18,}\{2, 6, 10, 14, 18, \dots \} contains even numbers.
  • D) {3,8,12,16,20,}\{3, 8, 12, 16, 20, \dots \} includes odd and even numbers but does not start with 0.

Thus, option C is the correct answer because it contains all even numbers.

Would you like more details or have any further questions?

Related Questions:

  1. How do you solve linear congruences in general?
  2. What does the term "modulo" mean in modular arithmetic?
  3. How can you determine whether a congruence has a unique solution?
  4. What are some properties of even and odd numbers in modular arithmetic?
  5. Can you apply the same method to solve quadratic congruences?

Tip: When solving congruences, always try simplifying the equation step by step, and pay attention to patterns in modular systems like mod 4 or mod 5.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Linear Congruences

Formulas

2x + 3 ≡ 3 mod 4
x ≡ 0 mod 2

Theorems

Properties of Modular Arithmetic
Linear Congruence Theorem

Suitable Grade Level

Grades 9-12