The correct answer is:

A) GCD(a, n) | b

Explanation:

For the linear congruence $ax \equiv b \ (\text{mod} \ n)$ to have a solution, it is necessary and sufficient that the greatest common divisor (GCD) of $a$ and $n$ divides $b$. In other words, $\text{GCD}(a, n)$ must be a divisor of $b$.

This can be understood by noting that if $ax \equiv b \ (\text{mod} \ n)$, then there exists an integer $k$ such that:

$ax = b + kn$

This equation implies that $ax - kn = b$. Hence, $b$ must be divisible by the GCD of $a$ and $n$, which is why the condition $\text{GCD}(a, n) | b$ is necessary for the existence of a solution.

Would you like further details or have any questions on this topic?

Here are five related questions you might consider:

1. How do you solve a linear congruence equation when the GCD condition is satisfied?
2. Can you explain how the Chinese Remainder Theorem is applied to solve systems of linear congruences?
3. What happens if $\text{GCD}(a, n)$ does not divide $b$ in a linear congruence equation?
4. How is the concept of modular inverses related to solving linear congruences?
5. What is the method of successive substitutions in solving systems of congruences?

Tip: When dealing with linear congruences, always start by checking the GCD condition to determine whether a solution exists before attempting to find it.