Question

Show that every positive odd integer is of the form (4q+1) or (4q+2) for some integer q.

Answer 1

Let a be the given positive odd integer. 

On dividing a by 4, let q be the quotient and r the remainder. 

Therefore, by Euclid’s algorithm we have 

a = 4q + r 0 ≤ r ˂ 4 

⇒ a = 4q + \(\pi\)r = 0, 1, 2, 3 

⇒ a = 4q, a = 4q + 1, a = 4q + 2, a = 4q + 3 

But, 4q and 4q + 2 = 2 (2q + 1) = even 

Thus, when a is odd, it is of the form (4q + 1) or (4q + 3) for some integer q.