In this problem, we show the possible forms of any positive odd integer. We will prove that every positive odd integer can be expressed in one of the forms 6q + 1, 6q + 3, or 6q + 5, where q is an integer.

Question

Show that any positive odd integer is of the form
$6q + 1$ or $6q + 3$ or $6q + 5$,
where q is some integer.

Solution

Let n be any positive integer.

Every positive integer can be written in one of the following six forms:

$$6q,\; 6q+1,\; 6q+2,\; 6q+3,\; 6q+4,\; 6q+5$$

where q is an integer.

Now, we examine which of these forms represent odd integers.

• $6q$ is even

• $6q+2$ is even

• $6q+4$ is even

So these forms cannot represent odd integers.

The remaining forms are:

$$6q+1,\; 6q+3,\; 6q+5$$

Each of these numbers is odd.

Hence, any positive odd integer must be of the form

$$6q + 1 \quad \text{or} \quad 6q + 3 \quad \text{or} \quad 6q + 5$$

where q is an integer.

Conclusion

Therefore, every positive odd integer can be written in the form
$6q + 1$, $6q + 3$, or $6q + 5$, where q is some integer.

$$\boxed{\text{Hence proved.}}$$

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