Which of the following equations has no solution in modular arithmetic?

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Question: Which of the following equations has no solution in modular arithmetic?

Options:

Correct Answer: 3x ≡ 9 (mod 6)

Solution:

3x ≡ 9 (mod 6) has no solution because 3 and 6 are not coprime, and 9 is not divisible by the gcd(3, 6).

Which of the following equations has no solution in modular arithmetic?

Which of the following equations has no solution in modular arithmetic?

Step 1: Understand what modular arithmetic means. It involves equations where numbers wrap around after reaching a certain value (the modulus).
Step 2: Identify the equation given: 3x ≡ 9 (mod 6). This means we are looking for values of x that make 3x equal to 9 when considered under modulo 6.
Step 3: Find the greatest common divisor (gcd) of the coefficient of x (which is 3) and the modulus (which is 6). The gcd(3, 6) is 3.
Step 4: Check if the right side of the equation (which is 9) is divisible by the gcd we found (which is 3). Since 9 ÷ 3 = 3, it is divisible.
Step 5: Now, check if the coefficient of x (3) and the modulus (6) are coprime. Two numbers are coprime if their gcd is 1. Since gcd(3, 6) is 3, they are not coprime.
Step 6: Since the coefficient and modulus are not coprime, and the right side is not divisible by the gcd, the equation has no solution.

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