Question

If $(3 - x) \equiv (2x - 5) \pmod{4}$, then one of the values of $x$ is

• A. 3
• B. 4
• C. 18
• D. 5

Hint:

Convert the congruence to a divisibility condition: $4 \mid (a - b)$, then test integer values of $k$ to find integer solutions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
$a \equiv b \pmod{m}$ means $m \mid (a - b)$.
Step 2: Detailed Explanation:
$(3-x) - (2x-5) = 8 - 3x$ must be divisible by 4.
$8 - 3x = 4k \Rightarrow 3x = 8 - 4k$.
For $k = -1$: $3x = 12 \Rightarrow x = 4$. Verify: $3 - 4 = -1$, $2(4)-5 = 3$; $(-1) - 3 = -4 \equiv 0 \pmod{4}$. \checkmark
Step 3: Final Answer:
One value of $x$ is $4$.