Which of the following sets of numbers can be expressed in the form 4q + 1 or 4q + 3, where q is an integer?

Let a be any positive odd integer and let b = 4. 

Then, by Euclid’s division lemma, integers q and r exist such that 

a = 4q + r, 0 ≤ r < 4 

Then, r = 0 or 1 or 2 or 3. 

Thus, a can be 4q or 4q + 1 or 4q + 2 or 4q + 3. 

Since a is odd, a cannot be 4q or 4q + 2, as they are divisible by 2. 

Thus, a can be expressed in the form 4q + 1 or 4q + 3. 

Hence, every positive odd integer can be expressed in the form 4q + 1 or 4q + 3, where q is an integer.