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