The solution of a linear congruence ax=b(modn) exists if

A) GCD (a, n) | b

B) GCD (a, b) In

C) GCD (a, n) = 1

D) GCD (a, b) =1

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.

Explore the conditions under which a linear congruence ax ≡ b (mod n) has a solution. Learn about the importance of GCD (a, n) | b in determining the existence of solutions. Suitable for advanced high school students and those interested in number theory.

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