An odd integer $n$ can be represented as $n=2k + 1$, where $k$ is an integer.

If $k$ is divided by $3$, then the possible remainders are $0,~ 1,~ 2$. So, the possible forms of $k$ are $k=3q$, $k=3q+1$ or $k=3q+2$, where $q$ is an integer.

Thus, we have the following casses:

1. When $k$ is of the form $3q$, then $n$ takes the following form:
$$n=2k + 1 = 2(3q) + 1 = 6q + 1.$$

2. When $k$ is of the form $3q+1$, then $n$ takes the following form:
$$n=2k + 1 = 2(3q + 1) + 1 = 6q + 2 + 1 = 6q + 3.$$

3. When $k$ is of the form $3q$, then $n$ takes the following form:
$$n=2k + 1 = 2(3q + 2) + 1 = 6q + 4 + 1 = 6q + 5.$$

Therefore, any odd integer $n$ can be expressed as $6q + 1$, $6q + 3$ or $6q + 5$, where $q$ is an integer.