Suppose that the integers $a < b < c < n$ have $a + b + c + n = 100$. Since $a < b < c < n$, then $a + b + c + n < n + n + n + n = 4n$. Thus, $100 < 4n$ and so $n > 25$. Since $n$ is an integer, then $n$ is at least 26. Could $n$ be 26? In this case, we would have $a + b + c = 100 - 26 = 74$. If $n = 26$, then $a + b + c$ is at most $23 + 24 + 25 = 72$, which means that we cannot have $a + b + c = 74$. Therefore, $n$ cannot be 26. Could $n$ be 27? In this case, we would have $a + b + c = 100 - 27 = 73$. Here, we could have $a + b + c = 23 + 24 + 26 = 73$, and so $n = 27$ is possible, which means that the smallest possible value of $n$ is 27.