Step 1: Understanding the Concept:
This question is about solving a linear congruence of the form \(ax \equiv b \pmod{n}\). The number of solutions to such a congruence depends on the greatest common divisor (GCD) of the coefficient \(a\) and the modulus \(n\).
Step 2: Key Formula or Approach:
A linear congruence \(ax \equiv b \pmod{n}\) has solutions if and only if \(\gcd(a, n)\) divides \(b\).
Let \(d = \gcd(a, n)\).
- If \(d\) does not divide \(b\), there are no solutions.
- If \(d\) divides \(b\), there are exactly \(d\) incongruent solutions modulo \(n\).
Step 3: Detailed Explanation:
The given congruence is \(10x \equiv 15 \pmod{35}\).
Here, we have \(a = 10\), \(b = 15\), and \(n = 35\).
First, we calculate the greatest common divisor of \(a\) and \(n\):
\[ d = \gcd(10, 35) \]
The divisors of 10 are {1, 2, 5, 10}.
The divisors of 35 are {1, 5, 7, 35}.
The greatest common divisor is 5. So, \(d = 5\).
Next, we check if \(d\) divides \(b\). We need to check if 5 divides 15.
\[ 15 \div 5 = 3 \]
Yes, 5 divides 15.
Since \(\gcd(10, 35)\) divides 15, the congruence has solutions.
The number of incongruent solutions modulo 35 is equal to \(d\).
Therefore, there are exactly 5 solutions.
Step 4: Final Answer:
The congruence has \(\gcd(10, 35) = 5\) solutions. This matches option (C).