**2. 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 **a** be any positive odd integer and **b=6**.

We can apply **Euclid's division algorithm** on **a **and **b=6**.

We know that value of .

Therefore, all possible values of **a** are:

a=6q

a=6q+1

a=6q+2

a=6q+3

a=6q+4

a=6q+5

We can ignore **6q, 6q+2 and 6q+4** because they are divisible by 2 which means they are not positive odd integers.

Therefore, we are just left with **6q+1, 6q+3 and 6q+5**.

Therefore, any positive odd integer is of the form **(6q+1) or (6q+3)or (6q+5)**.