**Solution:**

Let a be any positive integer and b = 6.

Then, by Euclid’s algorithm, a = 6q + r for

some integer q ≥ 0 and r = 0,1,2,3,4,5 because

0 ≤ r ≤ 6.

So, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

Here, a cannot be 6q or 6q + 2 or 6q + 4, as they are divisible by 2.

6q + 1

6 is divisible by 2 but 1 is not divisible by 2.

6q + 3

6 is divisible by 2 but 3 is not divisible by 2.

6q + 5

6 is divisible by 2 but 5 is not divisible by 2.

Since, 6q + 1, 6q + 3, 6q + 5 are not divisible by 2, they are odd numbers.

Therefore, any odd integer is of the form

6q + 1, or 6q + 3, or 6q + 5.