Let $n$ be an integer. Then the sum of three consecutive integers is $n + (n + 1) + (n + 2) = 3n + 3$,
which is always divisible by 3. However, we are looking for a prime number that is a factor of every sum of three consecutive integers.
The only prime number that divides every multiple of 3 is $\boxed{3}$.
The answer is: 3
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the prime number that is a factor of every sum of three consecutive integers?

### Answer:
