Let n be a positive odd integer. We need to show that n can be written in any one of the form of 8q+1, 8q+3, 8q+5 or 8q+7

According to division algorithm,
we can write any number ‘a’ in the form
a = 8q + r
where q is any integer and 0 <= r <= 7. So r can be 0, 1, 2, 3, 4, 5, 6 or 7.

Thus, a can be written as
a = 8q
a = 8q+2
a = 8q+3
a = 8q+4
a = 8q+5
a = 8q+6
a = 8q+7

We need only odd numbers. Since 8q, 8q+2, 8q+4, and 8q+6 are divisible by 2, they are even numbers.

So any odd integer can be written as any one of the remaining forms which are (8q+1, 8q+3, 8q+5 or 8q+7.)