The incongruent solution of 5x=2(mod4) is

A) 10

B) 6

C) 1

D) 2

To find the incongruent solution of the congruence \(5x = 2 \pmod{4}\), we start by simplifying and analyzing the congruence.

### Step 1: Simplify the congruence
The congruence \(5x = 2 \pmod{4}\) can be rewritten as:

\[
5x \equiv 2 \pmod{4}
\]

However, we note that \(5 \equiv 1 \pmod{4}\). Thus, this simplifies to:

\[
x \equiv 2 \pmod{4}
\]

### Step 2: Identify the incongruent solution
We are looking for the solution that is **not congruent** to \(x \equiv 2 \pmod{4}\). This means we are looking for a number that does not satisfy \(x = 4k + 2\) for any integer \(k\).

Let’s examine the given options:
- A) 10: \(10 \equiv 2 \pmod{4}\) (congruent)
- B) 6: \(6 \equiv 2 \pmod{4}\) (congruent)
- C) 1: \(1 \equiv 1 \pmod{4}\) (incongruent)
- D) 2: \(2 \equiv 2 \pmod{4}\) (congruent)

The incongruent option is:

**C) 1**

Would you like a more detailed explanation or have any other questions?

Here are some related questions to explore:
1. How to solve linear congruences in general?
2. What does it mean for two numbers to be congruent modulo \(n\)?
3. How to find all solutions to \(ax \equiv b \pmod{n}\)?
4. How can you determine if a congruence has no solution?
5. How to handle modular arithmetic with negative numbers?

**Tip:** When working with congruences, simplifying coefficients by reducing them modulo \(n\) can often make the problem much easier to solve.

Learn how to find the incongruent solution of the congruence 5x ≡ 2 (mod 4) with a step-by-step explanation. This problem involves modular arithmetic and linear congruences, exploring properties and solutions. Suitable for high school students.

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