If 3x ≡ 9 (mod 12), what is the value of x?

If 3x ≡ 9 (mod 12), what is the value of x?

Step 1: Start with the equation 3x ≡ 9 (mod 12).
Step 2: Understand that 'mod 12' means we are working with numbers that wrap around after 12.
Step 3: To simplify the equation, we can divide both sides by 3. This is allowed because 3 has a multiplicative inverse modulo 12.
Step 4: Dividing both sides gives us x ≡ 3 (mod 12).
Step 5: This means that x can be 3, or any number that is 3 more than a multiple of 12 (like 3, 15, 27, etc.).
Step 6: The simplest solution is x = 3.

Modular Arithmetic – Understanding how to solve equations in the context of modular arithmetic, including the properties of congruences.
Divisibility – Recognizing when it is valid to divide both sides of a congruence by a number, particularly in relation to the modulus.